Research ArticleCategories of (119868 119868)-Fuzzy Greedoids

Zhen-Yu Xiu1 and Fu-Gui Shi2

1College of Applied Mathematics Chengdu University of Information Technology Chengdu 610000 China2School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Zhen-Yu Xiu xyz198202163com

Received 3 July 2015 Accepted 18 October 2015

Academic Editor Pasquale Vetro

Copyright copy 2015 Z-Y Xiu and F-G Shi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The concepts of 119868-greedoids fuzzifying greedoids and (119868 119868)-fuzzy greedoids are introduced and feasibility preserving mappingsbetween greedoids are definedThen 119868-feasibility preserving mappings fuzzifying feasibility preserving mappings and (119868 119868)-fuzzyfeasibility preserving mappings are given as generalizations of feasibility preserving mappings We study the relations amonggreedoids 119868-greedoids fuzzifying greedoids and (119868 119868)-fuzzy greedoids from a categorical point of view

1 Introduction

Greedoids have been invented by Korte and Lovasz in [1 2]Originally the main motivation for proposing this gener-alization of the matroid concept came from combinatorialoptimization The optimality of the greedy algorithm couldin several instances be traced back to be an underlying com-binatorial structure that was not a matroid but a greedoidOptimality of the greedy solution for a broad class of objectivemappings characterizes these structures Many algorithmicapproaches in different areas of combinatorics and otherfields of numericalmathematics define the structure of a gree-doid Examples are scheduling under precedence constraintsbreadth first search shortest path Gaussian eliminationshellings of trees chordal graphs and convex sets line andpoint search series-parallel decomposition retracting anddismantling of posets and graphs and bisimplicial elimina-tion

The fuzzification of matroids was first investigated byGoetschel and Voxman [3] and the concept of fuzzy matroidswas introduced where a family of independent fuzzy setswas defined as a crisp family of fuzzy subsets of a finite setsatisfying certain set of axioms Subsequently many authorsinvestigated Goetschel-Voxman fuzzy matroids (see [3ndash12])The concept of 119872-fuzzifying matroids was introduced as

a new approach to the fuzzification of matroids by Shi [13]and his approach to the fuzzification of matroids preservesmany basic properties of crisp matroids (see [14ndash17]) Par-ticularly the categorical relations among matroids fuzzymatroids and fuzzifying matroids are studied [18] and themain results are shown as follows

M119903119888

sub FYM cong CFM119888

sub FM (1)

where 119903 119888 in the diagram mean respectively reflective andcoreflective

In [19] the concepts of 119871-matroids and (119871119872)-fuzzymatroids are introduced as generations of matroids and werewidely investigated (see [20ndash23]) In [22] the relations amongmatroids [0 1]-prematroids fuzzifying matroids and 119887119894-fuzzy prematroids were studied from a categorical viewpointThe main results are summarized as follows

M119903119888

sub IM

FYM119903119888

sub IIFM

M119903119888

sub FYM cong CPIM119888

sub IM119903119888

sub IIFM

(2)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 543247 10 pageshttpdxdoiorg1011552015543247

2 Journal of Function Spaces

where 119903 119888 in the diagram mean respectively reflective andcoreflective In order to see the relations clearly we give thefollowing equivalent diagram

r c

r c

c iso FYM

r c

r cM

CPIMIM

IIFM

(3)

The aim of this paper is to introduce the concepts of119868-greedoids fuzzifying greedoids and (119868 119868)-fuzzy greedoidsand study the relations among greedoids 119868-greedoids fuzzi-fying greedoids and (119868 119868)-fuzzy greedoids from a categoricalpoint of view It is easy to prove that greedoids and feasibilitypreserving mappings form a category fuzzifying greedoidsand fuzzifying feasibility preserving mappings form a cate-gory 119868-greedoids and 119868-feasibility preserving mappings forma category and (119868 119868)-fuzzy greedoids and (119868 119868)-fuzzy feasi-bility preserving mappings form a category In what followsthey are denoted byGFYG IG and IIFG respectivelyCPIGdenotes the category of closed and perfect 119868-greedoids and 119868-feasibility preserving mappings as morphisms

The paper is organized as follows In Sections 3 and 4the concepts of 119868-greedoids fuzzifying greedoids and (119868 119868)-fuzzy greedoids are introduced respectively In Section 5 weshow that FYG is isomorphic to CPIG and is a concretelycoreflective full subcategory of IG In Section 6 we show thatG can be embedded in FYG as a simultaneously concretelyreflective and coreflective full subcategory and IG is asimultaneously concretely reflective and coreflective full sub-category of IIFG In Section 7G can be embedded in IG as aconcretely coreflective full subcategory andFYG is a reflectivefull subcategory of IIFG In summary we show that

IIFG

IG

r c

r c

CPIGc iso FYG

c

Gc

(4)

where 119903 119888 in the diagram mean respectively reflective andcoreflective

2 Preliminaries

Throughout this paper 119868 = [0 1] and 119864 is a nonempty finiteset We denote the set of all subsets of 119864 by 2119864 and the set ofall fuzzy subsets of 119864 by 119868119864

For 119860 119861 sube 119864 119860 minus 119861 = 119890 isin 119864 119890 isin 119860 119890 notin 119861

For 119886 isin (0 1] and 119860 sube 119864 define a fuzzy set 119886 and 119860 asfollows

(119886 and 119860) (119890) =

119886 119890 isin 119860

0 119890 notin 119860

(5)

A fuzzy set 119886 and 119890 is called a fuzzy point and denoted by 119890119886

For 119886 isin (0 1] and 119860 isin 119868119864 we define

119860[119886]= 119890 isin 119864 119860 (119890) ge 119886

119860(119886)= 119890 isin 119864 119860 (119890) gt 119886

(6)

Definition 1 (see [24]) IfI is a nonempty subset of 2119864 thenthe pair (119864I) is called a (crisp) set system A set system(119864I) is called a matroid if it satisfies the following condi-tions

(I1) 0 isin I(I2) If 119860 isin I 119861 isin 2119864 and 119861 sube 119860 then 119861 isin I(I3) If119860 119861 isin I and |119860| lt |119861| then there is 119890 isin 119861minus119860 such

that 119860 cup 119890 isin I

Definition 2 (see [25]) A greedoid is a pair of (119864F) whereF sube 2

119864 is a set system satisfying the following conditions

(G1) For every 119860 isin F there is an 119890 isin 119860 such that 119860 minus 119890 isinF

(G2) For119860 119861 isin 2119864 such that |119860| lt |119861| there is an 119890 isin 119861minus119860such that 119860 cup 119890 isin F

The sets inF are called feasible (rather than ldquoindependentrdquo)

Remark 3 In [25] (G11015840) and (G2) together define greedoidsas well (G1) and (G2) where (G11015840) 0 isin F Obviously (G1)in Definition 2 could be replaced by the weaker axiom (G11015840)and greedoids are defined as generalizations of matroids

Definition 4 Let (119864119894F119894) (119894 = 1 2) be greedoids A mapping

119891 1198641rarr 1198642is called a feasibility preserving mapping from

(1198641F1) to (119864

2F2) if 119891minus1(119860) isin F

1for all 119860 isin F

2

Remark 5 In [26] a function between two convex structurescalled a convexity preserving function inverts convex setsinto convex sets Here similarly we give a mapping betweentwo greedoids which inverts feasible sets into feasible setsand is called a feasibility preserving mapping

Definition 6 (see [13]) A mapping I 2119864

rarr 119872 is calledan119872-fuzzy family of independent sets on 119864 if it satisfies thefollowing conditions

(FI1) I(0) = ⊤119872

(FI2) For any 119860 119861 isin 2119864 119860 supe 119861 rArr I(119860) ⩽ I(119861)(FI3) For any 119860 119861 isin 2119864 if |119860| lt |119861| then ⋁

119890isin119861minus119860I(119860 cup

119890) ⩾ I(119860) andI(119861)

IfI is an119872-fuzzy family of independent sets on 119864 then thepair (119864I) is called an 119872-fuzzifying matroid For 119860 isin 2119864I(119860) can be regarded as the degree to which 119860 is an inde-pendent set

Journal of Function Spaces 3

Definition 7 (see [19]) A subfamilyI of 119871119864 is called a familyof independent 119871-fuzzy sets on 119864 if it satisfies the followingconditions

(LI1) 1205940isin I

(LI2) 119860 isin 119871119864 119861 isin I and 119860 sube 119861 rArr 119860 isin I(LI3) If 119860 119861 isin I and 119887 = |119861|(119899) 997408 |119860|(119899) for some 119899 isin N

then there exists 119890 isin 119865(119860 119861) such that (119887and119860[119887])cup119890119887isin

I where 119865(119860 119861) = 119890 isin 119864 119887 ⩽ 119861(119890) 119887 997408 119860(119890)

If I is a family of independent 119871-fuzzy sets on 119864 then thepair (119864I) is called an 119871-matroid

Definition 8 (see [19]) A mappingI 119871119864

rarr 119872 is called an119872-fuzzy family of independent 119871-fuzzy sets on 119864 if it satisfiesthe following conditions

(LMFI1)I(1205940) = ⊤119872

(LMFI2) For any 119860 119861 isin 119871119864 119860 sube 119861 rArr I(119860) ⩾ I(119861)(LMFI3) If 119887 = |119861|(119899) 997408 |119860|(119899) for 119860 119861 isin 119871119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

I ((119887 and 119860[119887]) cup 119890119887) ⩾ I (119860) andI (119861) (7)

IfI is an119872-fuzzy family of independent 119871-fuzzy sets on 119864then the pair (119864I) is called an (119871119872)-fuzzy matroid

Remark 9 (1) In [13 19]119871 and119872 denote completely distribu-tive lattices In [22] when 119871 = 119872 = [0 1] an (119871119872)-fuzzymatroid is also called 119887119894-fuzzy prematroid an 119871-matroidis called [0 1]-prematroid and an 119872-fuzzifying matroid iscalled a fuzzifying matroid

(2) When 119871 is replaced by the interval [0 1] it is easy tosee 119869(119871) = 119871 perp

119871 = (0 1]

Definition 10 (see [19]) Let 119860 be an 119871-fuzzy set on a finiteset 119864 Then the mapping |119860| N rarr 119871 defined by forall119899 isin N|119860|(119899) = ⋁119886 isin 119871 |119860

[119886]| ge 119899 is called the 119871-fuzzy

cardinality of 119860

Lemma 11 (see [19]) For a finite set 119864 it holds that |119860|[119886]=

|119860[119886]| for any 119860 isin 119871119864 and any 119886 isin 119869(119871)

Lemma 12 (see [19]) Let119866 sube 119864Then |119887and119866|(119899) = (119887and|119866|)(119899)for any 119899 gt 0 and for any 119887 isin 119871 perp

119871

3 119868-Greedoids

Based onDefinitions 2 and 7 we give the following definition

Definition 13 AsubfamilyF of 119868119864 is called a family of feasiblefuzzy sets on 119864 if it satisfies the following conditions

(IG1) 1205940isin F

(IG2) If 119860 119861 isin F and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin Nthen there exists 119890 isin 119865(119860 119861) such that (119887and119860

[119887])cup119890119887isin

F where

119865 (119860 119861) = 119890 isin 119864 119861 (119890) ge 119887 gt 119860 (119890) (8)

IfF is a family of feasible fuzzy sets on119864 then the pair (119864F)is called an 119868-greedoid

Theorem 14 Let F sube 119868119864 forall119886 isin (0 1] define F[119886] = 119860

[119886]

119860 isin F If (119864F) is an 119868-greedoid then (119864F[119886]) is a greedoidfor any 119886 isin (0 1]

Proof Suppose that (119864F) is an 119868-greedoid Now we provethat forall119886 isin (0 1] (119864F[119886]) is a greedoid

(1) Obviously 0 = (1205940)[119886]isin F[119886] for any 119886 isin (0 1]

(2) If119860 119861 isin F[119886] and |119860| lt |119861| then there are119866119867 isin Fsuch that119860 = 119866

[119886] 119861 = 119867

[119886] and |119866

[119886]| lt |119867

[119886]| Next

we prove 119886 and 119866[119886] 119886 and 119867

[119886]isin F

If 119866[119886]= 0 then it is obviously 119886 and 119866

[119886]isin F

If |119866[119886]| = 119899 ge 1 let119866

[119886]= 1198901 1198902 119890119899 Since 120594

0 119866 isin F

it is easy to see that there exists 1198991isin N such that |119866|(119899

1) ge

119886 gt |1205940|(1198991) By (IG2) there exists 119890

1isin 119865(120594

0 119866) such that

(119886 and (1205940)[119886]) cup (1198901)119886isin F where 119865(120594

0 119866) = 119890 isin 119864 119866(119890) ge

119886 gt 1205940(119890) We get (119890

1)119886isin F and 119890

1isin 119866[119886]

Since (1198901)119886 119866 isin F it is easy to see that there exists 119899

2isin N

such that |119866|(1198992) ge 119886 gt |(119890

1)119886|(1198992) By (IG2) there exists

1198902isin 119865((119890

1)119886 119866) such that (119886 and ((119890

1)119886)[119886]) cup (1198902)119886isin F where

119865((1198901)119886 119866) = 119890 isin 119864 119866(119890) ge 119886 gt (119890

1)119886(119890) We get 119886 and

1198901 1198902 isin F and 119890

1 1198902 sube 119866[119886]

Continue the above process and we can obtain 119886 and 119866[119886]isin

FAnalogously we can obtain 119886 and 119867

[119886]isin F

Since |119866[119886]| lt |119867

[119886]| take 119899 isin N such that |119866

[119886]| lt 119899 le

|119867[119886]|Then by Lemma 12 |119886and119866

[119886]|(119899) = (119886and|119866

[119886]|)(119899) = 0 lt

119886 = (119886and|119867[119886]|)(119899) = |119886and119867

[119886]|(119899) Since 119886and119866

[119886] 119886and119867

[119886]isin F

there is 119890 isin 119864 such that (119886 and 119867[119886])(119890) gt (119886 and 119866

[119886])(119890) and

(119886 and (119886 and 119866[119886])[119886]) cup 119890119886isin F It is easy to see that 119866

[119886](119890) = 0

119867[119886](119890) = 1 and (119886 and 119866

[119886]) cup 119890119886= (119886 and (119886 and 119866

[119886])[119886]) cup 119890119886isin F

This implies that 119890 isin 119867[119886]= 119861 119890 notin 119866

[119886]= 119860 and 119860 cup 119890 =

119866[119886]cup 119890 = ((119886 and 119866

[119886])[119886]cup 119890119886)[119886]isin F[119886]

Therefore (119864F[119886]) is a greedoid for any 119886 isin (0 1]

Corollary 15 Let (119864F) be an 119868-greedoid If 119860 isin F thenforall119886 isin (0 1] 119886 and 119860

[119886]isin F

Remark 16 If (119864F) is an [0 1]-prematroid it is obvious tosee that Corollary 15 holds by (LI2) An 119868-greedoid (119864F)need not satisfy (LI2) However by Theorem 14 Corollary 15still holds This is an important property for 119868-greedoidsMany results of 119868-greedoids in this paper are based on thisproperty

Definition 17 An 119868-greedoid (119864F) is called a perfect 119868-greedoid if it satisfies the following condition forall119860 isin 119868

119864 ifforall119886 isin (0 1] 119886 and 119860

[119886]isin F then 119860 isin F

ByCorollary 15 andDefinition 17 we can easily obtain thefollowing corollary

Corollary 18 An 119868-greedoid (119864F) is a perfect 119868-greedoid ifit satisfies the following condition forall119860 isin 119868

119864 if forall119886 isin (0 1]119860[119886]isin F[119886] then 119860 isin F

4 Journal of Function Spaces

Example 19 Let 119864 = 3 4 5 Define 119860 isin 119868119864 by

119860 (119890) =

0 119890 = 3

1

3 119890 = 4

1

2 119890 = 5

(9)

andF sube 119868119864 byF = 119861 isin 119868

119864

119861 le 12 and 5 cup 119861 isin 119868119864

119861 le

13 and 3 4 5 minus 13 and 3 It is easy to verify that (119864F) isan 119868-greedoid but it is not perfect since 119886 and 119860

[119886]isin F for all

119886 isin (0 1] but 119860 notin F

Lemma 20 Let (119864F) be an 119868-greedoid If 0 lt 119886 le 119887 le 1thenF[119886] supe F[119887]

Proof Let 119860 isin F[119887] By the definition of F[119887] there exists119866 isin F such that 119860 = 119866

[119887] By Corollary 15 we have 119887 and 119860 =

119887 and 119866[119887]isin F Then 119860 = (119887 and 119860)

[119886]isin F[119886]

Theorem 21 Let (119864F) be an 119868-greedoid DefineFlowast = 119860 isin119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then (119864Flowast) is an 119868-greedoid

Moreover if (119864F) is perfect thenF = Flowast

Proof (1) Obviously (1205940)[119886]= 0 isin F[119886] for any 119886 isin (0 1]

Thus 1205940isin Flowast

(2) Let 119860 119861 isin Flowast and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isinN Then |119861|

[119887]gt |119860|[119887] By Lemma 11 we know that |119861

[119887]| gt

|119860[119887]| Since 119860

[119887] 119861[119887]isin F[119887] and (119864F[119887]) is a greedoid

there exists 119890 isin 119861[119887]minus 119860[119887]

such that 119860[119887]cup 119890 isin F[119887] In

this case ((119887 and119860[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F[119887] It is obvious

that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F[119887] sube F[119886] for every

119886 le 119887 and ((119887 and119860[119887]) cup 119890119887)[119886]= 0 isin F[119886] for every 119886 gt 119887 This

implies that (119887 and 119860[119887]) cup 119890119887isin Flowast

(3) It is obvious thatF supe Flowast Since (119864F) is perfect byCorollary 18F sube Flowast HenceF = Flowast

Corollary 22 Let (119864F) be an 119868-greedoid Then (119864F) isperfect if and only ifF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886]

Definition 23 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids A map-

ping 119891 1198641rarr 1198642is called an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) if119891larr119868(119860) isin F

1for all119860 isin F

2

where 119891larr119868(119860) = 119860 ∘ 119891

Theorem 24 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids and let

119891 1198641rarr 119864

2be an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) If (119864

1F1) is a perfect 119868-greedoid then

the following conditions are equivalent

(1) 119891 is an 119868-feasibility preserving mapping from (1198641F1)

to (1198642F2)

(2) 119891 is a feasibility preserving mapping from (1198641F1[119886])

to (1198642F2[119886]) for each 119886 isin (0 1]

Proof (1) rArr (2) Let 119860 isin F2[119886] Then there exists 119860 isin F

2

such that 119860 = 119860[119886]isin F2[119886] Since 119891 is an 119868-feasibility pre-

servingmapping from (1198641F1) to (119864

2F2)119891larr119868(119860) isin F

1We

have 119891larr119868(119860)[119886]= 119891minus1

(119860[119886]) = 119891

minus1

(119860) isin F1[119886] This implies

that 119891 is a feasibility preserving mapping from (1198641F1[119886]) to

(1198642F2[119886]) for each 119886 isin (0 1]

(1) lArr (2) Assume that 119891 is a feasibility preservingmapping from (119864

1F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

Let 119860 isin F2 Then 119860

[119886]isin F2[119886] for all 119886 isin (0 1] Thus

119891larr

119868(119860)[119886]= 119891minus1

(119860[119886]) isin F

1[119886] for all 119886 isin (0 1] Since

(1198641F1) is a perfect 119868-greedoid by Corollary 18 119891larr

119868(119860) isin

F1 Therefore 119891 is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2)

Theorem 25 Let (119864F) be an 119868-greedoid Then forall119886 isin (0 1](119864F[119886]) is a greedoid Since 119864 is a set finite there is at most afinite number of greedoids on 119864 Thus there is a finite sequence0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 such that

(1) if 119887 119888 isin (119886119894minus1 119886119894) (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 lt 119886

119894lt 119888 lt 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]The sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof We define an equivalence relation sim on (0 1] by 119886 sim119887 hArr F[119886] = F[119887] Since 119864 is a finite set the numberof greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is an

interval We only need to show that forall119886 119887 isin 119868119894with 119886 le 119887 if

119888 isin [119886 119887] then 119888 isin 119868119894 Since 119886 le 119888 le 119887 by Lemma 20 we know

that F[119886] supe F[119888] supe F[119887] As 119886 119887 isin 119868119894 F[119886] = F[119888] Thus

119888 isin 119868119894by the definition of 119868

119894 This implies that 119868

119894is an interval

Let 119886119894minus1= inf 119868

119894and 119886

119894= sup 119868

119894(119894 = 1 2 119899) Obviously

the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is the

fundamental sequence for (119864F)

Definition 26 An 119868-greedoid (119864F) with the fundamentalsequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called a

closed 119868-greedoid if whenever 119886119894minus1lt 119886 le 119886

119894(1 le 119894 le 119903) then

F[119886] = F[119886119894]

Theorem 27 Let (119864F) be an 119868-greedoid with the fundamen-tal sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 Then (119864F)

is a closed 119868-greedoid if and only if it satisfies the followingcondition

(C) forall119886 isin (0 1] and 119860 isin 2119864 if 119887 and 119860 isin F for all

0 lt 119887 lt 119886 then 119886 and 119860 isin F

Proof Suppose that (119864F) satisfies (C) forall119886 isin (119886119894minus1 119886119894) (119894 =

1 2 119899) then F[119886] supe F[119886119894] Let 119860 isin F[119886] for all 119886 isin

(119886119894minus1 119886119894) Then 119886and119860 isin F for all 119886 isin (119886

119894minus1 119886119894) thus 119887 and119860 isin F

for all 0 lt 119887 lt 119886119894 SinceF satisfies (C) 119886

119894and 119860 isin F Thus 119860 =

(119886119894and119860)[119886119894]isin F[119886

119894]This implies thatF[119886] sube F[119886

119894] for all 119886 isin

(119886119894minus1 119886119894) Therefore F[119886] = F[119886

119894] for all 119886 isin (119886

119894minus1 119886119894] that

is (119864F) is closed Conversely assume that (119864F) is a closed119868-greedoid Let 119886 isin (0 1] 119860 isin 2119864 and 119887 and 119860 isin F for all 0 lt119887 lt 119886 Since 119886 isin (0 1] 119886 isin (119886

119894minus1 119886119894] for some 119894 = 1 2 119899

Take 1198870isin (119886119894minus1 119886) sube (119886

119894minus1 119886119894] Then 119887

0and 119860 isin F and thus

119860 isin F[1198870] Since (119864F) is closed 119860 isin F[119886

119894] Hence 119886

119894and119860 isin

F By Corollary 15 we have 119886 and119860 = 119886 and (119886119894and119860)[119886]isin F This

means thatF satisfies (C)

Journal of Function Spaces 5

4 (119868 119868)-Fuzzy Greedoids andFuzzifying Greedoids

Definition 28 AmappingF 119868119864

rarr 119868 is called a fuzzy familyof feasible fuzzy sets on 119864 if it satisfies the following condi-tions

(IIFG1)F(1205940) = 1

(IIFG2) If |119861|(119899) ge 119887 gt |119860|(119899) for 119860 119861 isin 119868119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

F ((119887 and 119860[119887]) cup 119890119887) ⩾ F (119860) andF (119861) (10)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890)

IfF is a fuzzy family of feasible fuzzy sets on 119864 then the pair(119864F) is called an (119868 119868)-fuzzy greedoid

Definition 29 AmappingF 2119864

rarr 119868 is called a fuzzy familyof feasible sets on 119864 if it satisfies the following conditions

(FYG1)F(0) = 1

(FYG2) For any 119860 119861 isin 2119864 if |119860| lt |119861| then

⋁119890isin119861minus119860

F(119860 cup 119890) ⩾ F(119860) andF(119861)

IfF is a fuzzy family of feasible sets on119864 then the pair (119864F)is called a fuzzifying greedoid For 119860 isin 2

119864 F(119860) can beregarded as the degree to which 119860 is a feasible set

Theorem 30 Let F 119868119864

rarr 119868 be a mapping Then the fol-lowing conditions are equivalent

(1) (119864F) is an (119868 119868)-fuzzy greedoid

(2) For each 119886 isin (0 1] (119864F[119886]) is an 119868-greedoid

(3) For each 119886 isin [0 1) (119864F(119886)) is an 119868-greedoid

Proof (1) rArr (2) (IG1) It is obvious that 1205940isin F[119886]

for any119886 isin (0 1]

(IG2) If 119860 119861 isin F[119886]

and |119861|(119899) ge 119887 gt |119860|(119899) for some119899 isin N then by (IIFG2) we have ⋁

119890isin119865(119860119861)F((119887 and 119860

[119887]) cup

119890119887) ⩾ F(119860) and F(119861) Further by 119860 119861 isin F

[119886] we obtain

⋁119890isin119865(119860119861)

F((119887 and119860[119887]) cup 119890119887) ⩾ 119886 Then there exists 119890 isin 119865(119860 119861)

such thatF((119887and119860[119887])cup119890119887) ⩾ 119886 that is (119887and119860

[119887])cup119890119887isin F[119886]

This shows thatF[119886]

satisfies (IG2) Therefore (119864F[119886]) is an

119868-greedoid for each 119886 isin (0 1](2) rArr (1) (IIFG1) For any 119886 isin (0 1] 120594

0isin F[119886] We have

F(1205940) = 1

(IIFG2) Suppose that 119860 119861 isin [0 1]119864 and |119861|(119899) ge 119887 gt|119860|(119899) for some 119899 isin N In order to prove (IIFG2) take 119886 isin(0 1] andF(119860) andF(119861) ge 119886 ThenF(119860) ge 119886 andF(119861) ge 119886This implies 119860 119861 isin F

[119886] Hence by (IG2) there exists 119890 isin

119865(119860 119861) such that (119887 and 119860[119887]) cup 119890119887isin F[119886] This impliesF((119887 and

119860[119887])cup119890119887) ⩾ 119886 Furtherwehave⋁

119890isin119865(119860119861)F((119887and119860

[119887])cup119890119887) ⩾ 119886

By the arbitrariness of 119886 we obtain or119890isin119865(119860119861)

F((119887 and 119860[119887]) cup

119890119887) ⩾ F(119860) andF(119861)Analogously we can obtain (1) hArr (3)

Corollary 31 Let F 2119864

rarr 119868 be a map Then the followingconditions are equivalent

(1) (119864F) is a fuzzifying greedoid(2) (119864F

[119886]) is a greedoid for each 119886 isin (0 1]

(3) (119864F(119886)) is a greedoid for each 119886 isin [0 1)

Definition 32 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

A mapping 119891 1198641

rarr 1198642is called an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2) if

F1(119891larr

119868(119860)) ge F

2(119860) for all 119860 isin 1198681198642 where 119891larr

119868(119860) = 119860 ∘ 119891

Definition 33 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

A mapping 119891 1198641rarr 1198642is called a fuzzifying feasibility pre-

serving mapping from (1198641F1) to (119864

2F2) ifF

1(119891minus1

(119860)) ge

F2(119860) for all 119860 isin 21198642

Theorem 34 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

and let 119891 1198641rarr 1198642be an (119868 119868)-fuzzy feasibility preserving

mapping from (1198641F1) to (119864

2F2) Then the following condi-

tions are equivalent

(1) 119891 is an (119868 119868)-fuzzy feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)(119886)) to (119864

2 (F2)(119886)) for each 119886 isin [0 1)

Proof (1) rArr (2) Let119891 be an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1F1) to (119864

2F2) and 119886 isin (0 1] forall119860 isin

(F2)[119886] we have F

1(119891larr

119868(119860)) ge F

2(119860) ge 119886 and 119891larr

119868(119860) isin

(F1)[119886] In other words 119891 is a fuzzy feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin

(0 1](1) rArr (2) Next we need to prove that F

1(119891larr

119868(119860)) ge

F2(119860) for any 119860 isin 119868

119864 Let F2(119860) ge 119886 Then 119860 isin

(F2)[119886] Since 119891 is an 119868-feasibility preserving mapping from

(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1] 119891larr

119868(119860) isin

(F1)[119886] that is F

1(119891larr

119868(119860)) ge 119886 By the arbitrariness of 119886

we haveF1(119891larr

119868(119860)) ge F

2(119860) Therefore 119891 is an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2)

Analogously we can obtain (1) hArr (3)

Corollary 35 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

and let119891 1198641rarr 1198642be a fuzzifying feasibility preservingmap-

ping from (1198641F1) to (119864

2F2) Then the following conditions

are equivalent

(1) 119891 is a fuzzifying feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is a feasibility preserving mapping from (1198641 (F1)[119886])

to (1198642 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is a feasibility preserving mapping from (1198641 (F1)(119886))

to (1198642 (F2)(119886)) for each 119886 isin [0 1)

Theorem 36 Let (119864F) be a fuzzifying greedoid Then forall119886 isin(0 1] (119864F[119886]) is a greedoid Since 119864 is a finite set there is at

6 Journal of Function Spaces

most a finite number of greedoids that can be defined on119864Thusthere is a finite sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

such that

(1) if 119887 119888 isin (119886119894minus1 119886119894] (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 le 119886

119894lt 119888 le 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]

The sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof For 119886 119887 isin (0 1] we can easily check that if 119886 le 119887 thenF[119886]supe F[119887] We define an equivalence relation sim on (0 1]

by 119886 sim 119887 hArr F[119886]= F[119887] Since 119864 is a finite set the number

of greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is

a left open and right closed interval It is trivial that each119868119894(119894 = 1 2 119899) is an interval Suppose that 119886

119894= sup 119868

119894

and (119864F119894) is the corresponding cut greedoid with respect

to 119868119894 Then F

119894supe F[119886119894]

holds For all 119860 isin F119894 119860 isin F

[119903]

and F(119860) ge 119903 for all 119903 isin 119868119894 It follows that F(119860) ge 119886

119894and

119860 isin F[119886119894]

Thus F119894sube F[119886119894]

Hence F119894= F[119886119894]

and 119886119894isin 119868119894

Obviously the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is

the fundamental sequence for (119864F)

5 FYG as a Subcategory of IG

In this section we will study the relation between fuzzifyinggreedoids and fuzzy greedoids from the viewpoint of categorytheory

Theorem 37 Let (119864F) be a fuzzifying greedoid and 120590(F) =119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then

(1) 120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) (119864 120590(F)) is a closed and perfect 119868-greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860[119886]isin 120590(F)[119886] where

119860 isin 120590(F) Then 119860[119886]isin F[119886]

by the definition of 120590(F) Thismeans that120590(F)[119886] sube F

[119886](forall119886 isin (0 1]) Conversely assume

that 119860 isin F[119886] It is obvious that (119886 and 119860)

[119887]= 119860 isin F

[119886]sube

F[119887]

for every 119886 ge 119887 and (119886 and 119860)[119887]= 0 isin F

[119887]for every

119886 lt 119887 Hence 119886 and 119860 isin 120590(F) So 119860 = (119886 and 119860)[119886]isin 120590(F)[119886]

This implies that 120590(F)[119886] supe F[119886](forall119886 isin (0 1]) Therefore

120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) It is easy to see that 120590(F) satisfies (IG1) Nowwe provethat 120590(F) satisfies (IG2) Suppose that 119860 119861 isin 120590(F) and|119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|[119887] We

know that |119861[119887]| gt |119860

[119887]| Since119860

[119887] 119861[119887]isin F[119886]

and (119864F[119887])

is a greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup 119890 isin

F[119887] In this case ((119887 and 119860

[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F

[119887] It is

obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F

[119887]sube F[119886]

for every 119886 le 119887 and ((119887 and 119860[119887]) cup 119890119887)[119886]= 0 isin F

[119886]for every

119886 gt 119887 This implies that (119887 and 119860[119887]) cup 119890119887isin 120590(F)

By (1) and the definition of 120590(F) 120590(F) = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

120590(F)[119886] By Corollary 22 (119864 120590(F)) is a perfect 119868-greedoidByTheorem 36 (119864 120590(F)) is a closed 119868-greedoid

Theorem 38 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an 119868-feasibility preserving

mapping from (1198641 120590(F1)) to (119864

2 120590(F2))

Proof Since 119891 is a fuzzifying feasibility preserving mappingfrom (119864

1F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

by Corollary 35 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120590(F1)[119886]) to (119864

2 120590(F2)[119886]) for each

119886 isin (0 1] Since (1198641 120590(F1)) is perfect by Theorem 24 119891

is an 119868-feasibility preserving mapping from (1198641 120590(F1)) to

(1198642 120590(F2))

Theorem 39 Let (119864F) be a closed and perfect 119868-greedoidDefine a map 120575(F) 2119864 rarr 119868 by

120575 (F) (119860) = ⋁119886 isin (0 1] 119860 isinF [119886] (11)

Then

(1) 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) (119864 120575(F)) is a fuzzifying greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860 isin F[119886] Then120575(F)(119860) ge 119886 by the definition of 120575(F) We have119860 isin 120575(F)

[119886]

This means that 120575(F)[119886]supe F[119886] Conversely assume that

119860 isin 120575(F)[119886] that is 120575(F)(119860) ge 119886 Since (119864F) is a closed 119868-

greedoid there exists 119887 isin (0 1] such that 119887 ge 119886 and119860 isin F[119887]By 119887 ge 119886 F[119886] supe F[119887] This means that 120575(F)

[119886]sube F[119886]

Therefore 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) By (1) and Corollary 31 (119864 120575(F)) is a fuzzifyinggreedoid

Theorem 40 Let (119864119894F119894) (119894 = 1 2) be closed and perfect 119868-

greedoids If 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) then119891 is a fuzzifying feasibility

preserving mapping from (1198641 120575(F1)) to (119864

2 120575(F2))

Proof Since 119891 is an 119868-feasibility preserving mapping from(1198641F1) to (119864

2F2) 119891 is a feasibility preserving mapping

from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1] by

Theorem 24 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120575(F1)[119886]) to (119864

2 120575(F2)[119886]) for each 119886 isin

(0 1] By Corollary 35 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120575(F1)) to (119864

2 120575(F2))

Theorem 41 (1) If (119864F) is a fuzzifying greedoid then 120575 ∘120590(F) = F

(2) If (119864F) is a closed and perfect 119868-greedoid then 120590 ∘120575(F) = F

Proof (1) ByTheorem 37 (119864 120590(F)) is a closed and perfect 119868-greedoidThen 120575∘120590(F)

[119886]= 120590(F)[119886] = F

[119886]for all 119886 isin (0 1]

byTheorems 37 and 39 Hence 120575 ∘ 120590(F) = F(2) By Theorems 37 and 39 (119864 120590 ∘ 120575(F)) is a closed and

perfect 119868-greedoid Then 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1]119860[119886]

isin 120575(F)[119886] Since (119864F) is a closed and perfect 119868-

greedoid 120575(F)[119886]= F[119886] for all 119886 isin (0 1] by Theorem 39

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

where 119903 119888 in the diagram mean respectively reflective andcoreflective In order to see the relations clearly we give thefollowing equivalent diagram

r c

r c

c iso FYM

r c

r cM

CPIMIM

IIFM

(3)

The aim of this paper is to introduce the concepts of119868-greedoids fuzzifying greedoids and (119868 119868)-fuzzy greedoidsand study the relations among greedoids 119868-greedoids fuzzi-fying greedoids and (119868 119868)-fuzzy greedoids from a categoricalpoint of view It is easy to prove that greedoids and feasibilitypreserving mappings form a category fuzzifying greedoidsand fuzzifying feasibility preserving mappings form a cate-gory 119868-greedoids and 119868-feasibility preserving mappings forma category and (119868 119868)-fuzzy greedoids and (119868 119868)-fuzzy feasi-bility preserving mappings form a category In what followsthey are denoted byGFYG IG and IIFG respectivelyCPIGdenotes the category of closed and perfect 119868-greedoids and 119868-feasibility preserving mappings as morphisms

The paper is organized as follows In Sections 3 and 4the concepts of 119868-greedoids fuzzifying greedoids and (119868 119868)-fuzzy greedoids are introduced respectively In Section 5 weshow that FYG is isomorphic to CPIG and is a concretelycoreflective full subcategory of IG In Section 6 we show thatG can be embedded in FYG as a simultaneously concretelyreflective and coreflective full subcategory and IG is asimultaneously concretely reflective and coreflective full sub-category of IIFG In Section 7G can be embedded in IG as aconcretely coreflective full subcategory andFYG is a reflectivefull subcategory of IIFG In summary we show that

IIFG

IG

r c

r c

CPIGc iso FYG

c

Gc

(4)

where 119903 119888 in the diagram mean respectively reflective andcoreflective

2 Preliminaries

Throughout this paper 119868 = [0 1] and 119864 is a nonempty finiteset We denote the set of all subsets of 119864 by 2119864 and the set ofall fuzzy subsets of 119864 by 119868119864

For 119860 119861 sube 119864 119860 minus 119861 = 119890 isin 119864 119890 isin 119860 119890 notin 119861

For 119886 isin (0 1] and 119860 sube 119864 define a fuzzy set 119886 and 119860 asfollows

(119886 and 119860) (119890) =

119886 119890 isin 119860

0 119890 notin 119860

(5)

A fuzzy set 119886 and 119890 is called a fuzzy point and denoted by 119890119886

For 119886 isin (0 1] and 119860 isin 119868119864 we define

119860[119886]= 119890 isin 119864 119860 (119890) ge 119886

119860(119886)= 119890 isin 119864 119860 (119890) gt 119886

(6)

Definition 1 (see [24]) IfI is a nonempty subset of 2119864 thenthe pair (119864I) is called a (crisp) set system A set system(119864I) is called a matroid if it satisfies the following condi-tions

(I1) 0 isin I(I2) If 119860 isin I 119861 isin 2119864 and 119861 sube 119860 then 119861 isin I(I3) If119860 119861 isin I and |119860| lt |119861| then there is 119890 isin 119861minus119860 such

that 119860 cup 119890 isin I

Definition 2 (see [25]) A greedoid is a pair of (119864F) whereF sube 2

119864 is a set system satisfying the following conditions

(G1) For every 119860 isin F there is an 119890 isin 119860 such that 119860 minus 119890 isinF

(G2) For119860 119861 isin 2119864 such that |119860| lt |119861| there is an 119890 isin 119861minus119860such that 119860 cup 119890 isin F

The sets inF are called feasible (rather than ldquoindependentrdquo)

Remark 3 In [25] (G11015840) and (G2) together define greedoidsas well (G1) and (G2) where (G11015840) 0 isin F Obviously (G1)in Definition 2 could be replaced by the weaker axiom (G11015840)and greedoids are defined as generalizations of matroids

Definition 4 Let (119864119894F119894) (119894 = 1 2) be greedoids A mapping

119891 1198641rarr 1198642is called a feasibility preserving mapping from

(1198641F1) to (119864

2F2) if 119891minus1(119860) isin F

1for all 119860 isin F

2

Remark 5 In [26] a function between two convex structurescalled a convexity preserving function inverts convex setsinto convex sets Here similarly we give a mapping betweentwo greedoids which inverts feasible sets into feasible setsand is called a feasibility preserving mapping

Definition 6 (see [13]) A mapping I 2119864

rarr 119872 is calledan119872-fuzzy family of independent sets on 119864 if it satisfies thefollowing conditions

(FI1) I(0) = ⊤119872

(FI2) For any 119860 119861 isin 2119864 119860 supe 119861 rArr I(119860) ⩽ I(119861)(FI3) For any 119860 119861 isin 2119864 if |119860| lt |119861| then ⋁

119890isin119861minus119860I(119860 cup

119890) ⩾ I(119860) andI(119861)

IfI is an119872-fuzzy family of independent sets on 119864 then thepair (119864I) is called an 119872-fuzzifying matroid For 119860 isin 2119864I(119860) can be regarded as the degree to which 119860 is an inde-pendent set

Journal of Function Spaces 3

Definition 7 (see [19]) A subfamilyI of 119871119864 is called a familyof independent 119871-fuzzy sets on 119864 if it satisfies the followingconditions

(LI1) 1205940isin I

(LI2) 119860 isin 119871119864 119861 isin I and 119860 sube 119861 rArr 119860 isin I(LI3) If 119860 119861 isin I and 119887 = |119861|(119899) 997408 |119860|(119899) for some 119899 isin N

then there exists 119890 isin 119865(119860 119861) such that (119887and119860[119887])cup119890119887isin

I where 119865(119860 119861) = 119890 isin 119864 119887 ⩽ 119861(119890) 119887 997408 119860(119890)

If I is a family of independent 119871-fuzzy sets on 119864 then thepair (119864I) is called an 119871-matroid

Definition 8 (see [19]) A mappingI 119871119864

rarr 119872 is called an119872-fuzzy family of independent 119871-fuzzy sets on 119864 if it satisfiesthe following conditions

(LMFI1)I(1205940) = ⊤119872

(LMFI2) For any 119860 119861 isin 119871119864 119860 sube 119861 rArr I(119860) ⩾ I(119861)(LMFI3) If 119887 = |119861|(119899) 997408 |119860|(119899) for 119860 119861 isin 119871119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

I ((119887 and 119860[119887]) cup 119890119887) ⩾ I (119860) andI (119861) (7)

IfI is an119872-fuzzy family of independent 119871-fuzzy sets on 119864then the pair (119864I) is called an (119871119872)-fuzzy matroid

Remark 9 (1) In [13 19]119871 and119872 denote completely distribu-tive lattices In [22] when 119871 = 119872 = [0 1] an (119871119872)-fuzzymatroid is also called 119887119894-fuzzy prematroid an 119871-matroidis called [0 1]-prematroid and an 119872-fuzzifying matroid iscalled a fuzzifying matroid

(2) When 119871 is replaced by the interval [0 1] it is easy tosee 119869(119871) = 119871 perp

119871 = (0 1]

Definition 10 (see [19]) Let 119860 be an 119871-fuzzy set on a finiteset 119864 Then the mapping |119860| N rarr 119871 defined by forall119899 isin N|119860|(119899) = ⋁119886 isin 119871 |119860

[119886]| ge 119899 is called the 119871-fuzzy

cardinality of 119860

Lemma 11 (see [19]) For a finite set 119864 it holds that |119860|[119886]=

|119860[119886]| for any 119860 isin 119871119864 and any 119886 isin 119869(119871)

Lemma 12 (see [19]) Let119866 sube 119864Then |119887and119866|(119899) = (119887and|119866|)(119899)for any 119899 gt 0 and for any 119887 isin 119871 perp

119871

3 119868-Greedoids

Based onDefinitions 2 and 7 we give the following definition

Definition 13 AsubfamilyF of 119868119864 is called a family of feasiblefuzzy sets on 119864 if it satisfies the following conditions

(IG1) 1205940isin F

(IG2) If 119860 119861 isin F and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin Nthen there exists 119890 isin 119865(119860 119861) such that (119887and119860

[119887])cup119890119887isin

F where

119865 (119860 119861) = 119890 isin 119864 119861 (119890) ge 119887 gt 119860 (119890) (8)

IfF is a family of feasible fuzzy sets on119864 then the pair (119864F)is called an 119868-greedoid

Theorem 14 Let F sube 119868119864 forall119886 isin (0 1] define F[119886] = 119860

[119886]

119860 isin F If (119864F) is an 119868-greedoid then (119864F[119886]) is a greedoidfor any 119886 isin (0 1]

Proof Suppose that (119864F) is an 119868-greedoid Now we provethat forall119886 isin (0 1] (119864F[119886]) is a greedoid

(1) Obviously 0 = (1205940)[119886]isin F[119886] for any 119886 isin (0 1]

(2) If119860 119861 isin F[119886] and |119860| lt |119861| then there are119866119867 isin Fsuch that119860 = 119866

[119886] 119861 = 119867

[119886] and |119866

[119886]| lt |119867

[119886]| Next

we prove 119886 and 119866[119886] 119886 and 119867

[119886]isin F

If 119866[119886]= 0 then it is obviously 119886 and 119866

[119886]isin F

If |119866[119886]| = 119899 ge 1 let119866

[119886]= 1198901 1198902 119890119899 Since 120594

0 119866 isin F

it is easy to see that there exists 1198991isin N such that |119866|(119899

1) ge

119886 gt |1205940|(1198991) By (IG2) there exists 119890

1isin 119865(120594

0 119866) such that

(119886 and (1205940)[119886]) cup (1198901)119886isin F where 119865(120594

0 119866) = 119890 isin 119864 119866(119890) ge

119886 gt 1205940(119890) We get (119890

1)119886isin F and 119890

1isin 119866[119886]

Since (1198901)119886 119866 isin F it is easy to see that there exists 119899

2isin N

such that |119866|(1198992) ge 119886 gt |(119890

1)119886|(1198992) By (IG2) there exists

1198902isin 119865((119890

1)119886 119866) such that (119886 and ((119890

1)119886)[119886]) cup (1198902)119886isin F where

119865((1198901)119886 119866) = 119890 isin 119864 119866(119890) ge 119886 gt (119890

1)119886(119890) We get 119886 and

1198901 1198902 isin F and 119890

1 1198902 sube 119866[119886]

Continue the above process and we can obtain 119886 and 119866[119886]isin

FAnalogously we can obtain 119886 and 119867

[119886]isin F

Since |119866[119886]| lt |119867

[119886]| take 119899 isin N such that |119866

[119886]| lt 119899 le

|119867[119886]|Then by Lemma 12 |119886and119866

[119886]|(119899) = (119886and|119866

[119886]|)(119899) = 0 lt

119886 = (119886and|119867[119886]|)(119899) = |119886and119867

[119886]|(119899) Since 119886and119866

[119886] 119886and119867

[119886]isin F

there is 119890 isin 119864 such that (119886 and 119867[119886])(119890) gt (119886 and 119866

[119886])(119890) and

(119886 and (119886 and 119866[119886])[119886]) cup 119890119886isin F It is easy to see that 119866

[119886](119890) = 0

119867[119886](119890) = 1 and (119886 and 119866

[119886]) cup 119890119886= (119886 and (119886 and 119866

[119886])[119886]) cup 119890119886isin F

This implies that 119890 isin 119867[119886]= 119861 119890 notin 119866

[119886]= 119860 and 119860 cup 119890 =

119866[119886]cup 119890 = ((119886 and 119866

[119886])[119886]cup 119890119886)[119886]isin F[119886]

Therefore (119864F[119886]) is a greedoid for any 119886 isin (0 1]

Corollary 15 Let (119864F) be an 119868-greedoid If 119860 isin F thenforall119886 isin (0 1] 119886 and 119860

[119886]isin F

Remark 16 If (119864F) is an [0 1]-prematroid it is obvious tosee that Corollary 15 holds by (LI2) An 119868-greedoid (119864F)need not satisfy (LI2) However by Theorem 14 Corollary 15still holds This is an important property for 119868-greedoidsMany results of 119868-greedoids in this paper are based on thisproperty

Definition 17 An 119868-greedoid (119864F) is called a perfect 119868-greedoid if it satisfies the following condition forall119860 isin 119868

119864 ifforall119886 isin (0 1] 119886 and 119860

[119886]isin F then 119860 isin F

ByCorollary 15 andDefinition 17 we can easily obtain thefollowing corollary

Corollary 18 An 119868-greedoid (119864F) is a perfect 119868-greedoid ifit satisfies the following condition forall119860 isin 119868

119864 if forall119886 isin (0 1]119860[119886]isin F[119886] then 119860 isin F

4 Journal of Function Spaces

Example 19 Let 119864 = 3 4 5 Define 119860 isin 119868119864 by

119860 (119890) =

0 119890 = 3

1

3 119890 = 4

1

2 119890 = 5

(9)

andF sube 119868119864 byF = 119861 isin 119868

119864

119861 le 12 and 5 cup 119861 isin 119868119864

119861 le

13 and 3 4 5 minus 13 and 3 It is easy to verify that (119864F) isan 119868-greedoid but it is not perfect since 119886 and 119860

[119886]isin F for all

119886 isin (0 1] but 119860 notin F

Lemma 20 Let (119864F) be an 119868-greedoid If 0 lt 119886 le 119887 le 1thenF[119886] supe F[119887]

Proof Let 119860 isin F[119887] By the definition of F[119887] there exists119866 isin F such that 119860 = 119866

[119887] By Corollary 15 we have 119887 and 119860 =

119887 and 119866[119887]isin F Then 119860 = (119887 and 119860)

[119886]isin F[119886]

Theorem 21 Let (119864F) be an 119868-greedoid DefineFlowast = 119860 isin119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then (119864Flowast) is an 119868-greedoid

Moreover if (119864F) is perfect thenF = Flowast

Proof (1) Obviously (1205940)[119886]= 0 isin F[119886] for any 119886 isin (0 1]

Thus 1205940isin Flowast

(2) Let 119860 119861 isin Flowast and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isinN Then |119861|

[119887]gt |119860|[119887] By Lemma 11 we know that |119861

[119887]| gt

|119860[119887]| Since 119860

[119887] 119861[119887]isin F[119887] and (119864F[119887]) is a greedoid

there exists 119890 isin 119861[119887]minus 119860[119887]

such that 119860[119887]cup 119890 isin F[119887] In

this case ((119887 and119860[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F[119887] It is obvious

that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F[119887] sube F[119886] for every

119886 le 119887 and ((119887 and119860[119887]) cup 119890119887)[119886]= 0 isin F[119886] for every 119886 gt 119887 This

implies that (119887 and 119860[119887]) cup 119890119887isin Flowast

(3) It is obvious thatF supe Flowast Since (119864F) is perfect byCorollary 18F sube Flowast HenceF = Flowast

Corollary 22 Let (119864F) be an 119868-greedoid Then (119864F) isperfect if and only ifF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886]

Definition 23 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids A map-

ping 119891 1198641rarr 1198642is called an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) if119891larr119868(119860) isin F

1for all119860 isin F

2

where 119891larr119868(119860) = 119860 ∘ 119891

Theorem 24 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids and let

119891 1198641rarr 119864

2be an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) If (119864

1F1) is a perfect 119868-greedoid then

the following conditions are equivalent

(1) 119891 is an 119868-feasibility preserving mapping from (1198641F1)

to (1198642F2)

(2) 119891 is a feasibility preserving mapping from (1198641F1[119886])

to (1198642F2[119886]) for each 119886 isin (0 1]

Proof (1) rArr (2) Let 119860 isin F2[119886] Then there exists 119860 isin F

2

such that 119860 = 119860[119886]isin F2[119886] Since 119891 is an 119868-feasibility pre-

servingmapping from (1198641F1) to (119864

2F2)119891larr119868(119860) isin F

1We

have 119891larr119868(119860)[119886]= 119891minus1

(119860[119886]) = 119891

minus1

(119860) isin F1[119886] This implies

that 119891 is a feasibility preserving mapping from (1198641F1[119886]) to

(1198642F2[119886]) for each 119886 isin (0 1]

(1) lArr (2) Assume that 119891 is a feasibility preservingmapping from (119864

1F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

Let 119860 isin F2 Then 119860

[119886]isin F2[119886] for all 119886 isin (0 1] Thus

119891larr

119868(119860)[119886]= 119891minus1

(119860[119886]) isin F

1[119886] for all 119886 isin (0 1] Since

(1198641F1) is a perfect 119868-greedoid by Corollary 18 119891larr

119868(119860) isin

F1 Therefore 119891 is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2)

Theorem 25 Let (119864F) be an 119868-greedoid Then forall119886 isin (0 1](119864F[119886]) is a greedoid Since 119864 is a set finite there is at most afinite number of greedoids on 119864 Thus there is a finite sequence0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 such that

(1) if 119887 119888 isin (119886119894minus1 119886119894) (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 lt 119886

119894lt 119888 lt 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]The sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof We define an equivalence relation sim on (0 1] by 119886 sim119887 hArr F[119886] = F[119887] Since 119864 is a finite set the numberof greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is an

interval We only need to show that forall119886 119887 isin 119868119894with 119886 le 119887 if

119888 isin [119886 119887] then 119888 isin 119868119894 Since 119886 le 119888 le 119887 by Lemma 20 we know

that F[119886] supe F[119888] supe F[119887] As 119886 119887 isin 119868119894 F[119886] = F[119888] Thus

119888 isin 119868119894by the definition of 119868

119894 This implies that 119868

119894is an interval

Let 119886119894minus1= inf 119868

119894and 119886

119894= sup 119868

119894(119894 = 1 2 119899) Obviously

the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is the

fundamental sequence for (119864F)

Definition 26 An 119868-greedoid (119864F) with the fundamentalsequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called a

closed 119868-greedoid if whenever 119886119894minus1lt 119886 le 119886

119894(1 le 119894 le 119903) then

F[119886] = F[119886119894]

Theorem 27 Let (119864F) be an 119868-greedoid with the fundamen-tal sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 Then (119864F)

is a closed 119868-greedoid if and only if it satisfies the followingcondition

(C) forall119886 isin (0 1] and 119860 isin 2119864 if 119887 and 119860 isin F for all

0 lt 119887 lt 119886 then 119886 and 119860 isin F

Proof Suppose that (119864F) satisfies (C) forall119886 isin (119886119894minus1 119886119894) (119894 =

1 2 119899) then F[119886] supe F[119886119894] Let 119860 isin F[119886] for all 119886 isin

(119886119894minus1 119886119894) Then 119886and119860 isin F for all 119886 isin (119886

119894minus1 119886119894) thus 119887 and119860 isin F

for all 0 lt 119887 lt 119886119894 SinceF satisfies (C) 119886

119894and 119860 isin F Thus 119860 =

(119886119894and119860)[119886119894]isin F[119886

119894]This implies thatF[119886] sube F[119886

119894] for all 119886 isin

(119886119894minus1 119886119894) Therefore F[119886] = F[119886

119894] for all 119886 isin (119886

119894minus1 119886119894] that

is (119864F) is closed Conversely assume that (119864F) is a closed119868-greedoid Let 119886 isin (0 1] 119860 isin 2119864 and 119887 and 119860 isin F for all 0 lt119887 lt 119886 Since 119886 isin (0 1] 119886 isin (119886

119894minus1 119886119894] for some 119894 = 1 2 119899

Take 1198870isin (119886119894minus1 119886) sube (119886

119894minus1 119886119894] Then 119887

0and 119860 isin F and thus

119860 isin F[1198870] Since (119864F) is closed 119860 isin F[119886

119894] Hence 119886

119894and119860 isin

F By Corollary 15 we have 119886 and119860 = 119886 and (119886119894and119860)[119886]isin F This

means thatF satisfies (C)

Journal of Function Spaces 5

4 (119868 119868)-Fuzzy Greedoids andFuzzifying Greedoids

Definition 28 AmappingF 119868119864

rarr 119868 is called a fuzzy familyof feasible fuzzy sets on 119864 if it satisfies the following condi-tions

(IIFG1)F(1205940) = 1

(IIFG2) If |119861|(119899) ge 119887 gt |119860|(119899) for 119860 119861 isin 119868119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

F ((119887 and 119860[119887]) cup 119890119887) ⩾ F (119860) andF (119861) (10)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890)

IfF is a fuzzy family of feasible fuzzy sets on 119864 then the pair(119864F) is called an (119868 119868)-fuzzy greedoid

Definition 29 AmappingF 2119864

rarr 119868 is called a fuzzy familyof feasible sets on 119864 if it satisfies the following conditions

(FYG1)F(0) = 1

(FYG2) For any 119860 119861 isin 2119864 if |119860| lt |119861| then

⋁119890isin119861minus119860

F(119860 cup 119890) ⩾ F(119860) andF(119861)

IfF is a fuzzy family of feasible sets on119864 then the pair (119864F)is called a fuzzifying greedoid For 119860 isin 2

119864 F(119860) can beregarded as the degree to which 119860 is a feasible set

Theorem 30 Let F 119868119864

rarr 119868 be a mapping Then the fol-lowing conditions are equivalent

(1) (119864F) is an (119868 119868)-fuzzy greedoid

(2) For each 119886 isin (0 1] (119864F[119886]) is an 119868-greedoid

(3) For each 119886 isin [0 1) (119864F(119886)) is an 119868-greedoid

Proof (1) rArr (2) (IG1) It is obvious that 1205940isin F[119886]

for any119886 isin (0 1]

(IG2) If 119860 119861 isin F[119886]

and |119861|(119899) ge 119887 gt |119860|(119899) for some119899 isin N then by (IIFG2) we have ⋁

119890isin119865(119860119861)F((119887 and 119860

[119887]) cup

119890119887) ⩾ F(119860) and F(119861) Further by 119860 119861 isin F

[119886] we obtain

⋁119890isin119865(119860119861)

F((119887 and119860[119887]) cup 119890119887) ⩾ 119886 Then there exists 119890 isin 119865(119860 119861)

such thatF((119887and119860[119887])cup119890119887) ⩾ 119886 that is (119887and119860

[119887])cup119890119887isin F[119886]

This shows thatF[119886]

satisfies (IG2) Therefore (119864F[119886]) is an

119868-greedoid for each 119886 isin (0 1](2) rArr (1) (IIFG1) For any 119886 isin (0 1] 120594

0isin F[119886] We have

F(1205940) = 1

(IIFG2) Suppose that 119860 119861 isin [0 1]119864 and |119861|(119899) ge 119887 gt|119860|(119899) for some 119899 isin N In order to prove (IIFG2) take 119886 isin(0 1] andF(119860) andF(119861) ge 119886 ThenF(119860) ge 119886 andF(119861) ge 119886This implies 119860 119861 isin F

[119886] Hence by (IG2) there exists 119890 isin

119865(119860 119861) such that (119887 and 119860[119887]) cup 119890119887isin F[119886] This impliesF((119887 and

119860[119887])cup119890119887) ⩾ 119886 Furtherwehave⋁

119890isin119865(119860119861)F((119887and119860

[119887])cup119890119887) ⩾ 119886

By the arbitrariness of 119886 we obtain or119890isin119865(119860119861)

F((119887 and 119860[119887]) cup

119890119887) ⩾ F(119860) andF(119861)Analogously we can obtain (1) hArr (3)

Corollary 31 Let F 2119864

rarr 119868 be a map Then the followingconditions are equivalent

(1) (119864F) is a fuzzifying greedoid(2) (119864F

[119886]) is a greedoid for each 119886 isin (0 1]

(3) (119864F(119886)) is a greedoid for each 119886 isin [0 1)

Definition 32 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

A mapping 119891 1198641

rarr 1198642is called an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2) if

F1(119891larr

119868(119860)) ge F

2(119860) for all 119860 isin 1198681198642 where 119891larr

119868(119860) = 119860 ∘ 119891

Definition 33 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

A mapping 119891 1198641rarr 1198642is called a fuzzifying feasibility pre-

serving mapping from (1198641F1) to (119864

2F2) ifF

1(119891minus1

(119860)) ge

F2(119860) for all 119860 isin 21198642

Theorem 34 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

and let 119891 1198641rarr 1198642be an (119868 119868)-fuzzy feasibility preserving

mapping from (1198641F1) to (119864

2F2) Then the following condi-

tions are equivalent

(1) 119891 is an (119868 119868)-fuzzy feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)(119886)) to (119864

2 (F2)(119886)) for each 119886 isin [0 1)

Proof (1) rArr (2) Let119891 be an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1F1) to (119864

2F2) and 119886 isin (0 1] forall119860 isin

(F2)[119886] we have F

1(119891larr

119868(119860)) ge F

2(119860) ge 119886 and 119891larr

119868(119860) isin

(F1)[119886] In other words 119891 is a fuzzy feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin

(0 1](1) rArr (2) Next we need to prove that F

1(119891larr

119868(119860)) ge

F2(119860) for any 119860 isin 119868

119864 Let F2(119860) ge 119886 Then 119860 isin

(F2)[119886] Since 119891 is an 119868-feasibility preserving mapping from

(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1] 119891larr

119868(119860) isin

(F1)[119886] that is F

1(119891larr

119868(119860)) ge 119886 By the arbitrariness of 119886

we haveF1(119891larr

119868(119860)) ge F

2(119860) Therefore 119891 is an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2)

Analogously we can obtain (1) hArr (3)

Corollary 35 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

and let119891 1198641rarr 1198642be a fuzzifying feasibility preservingmap-

ping from (1198641F1) to (119864

2F2) Then the following conditions

are equivalent

(1) 119891 is a fuzzifying feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is a feasibility preserving mapping from (1198641 (F1)[119886])

to (1198642 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is a feasibility preserving mapping from (1198641 (F1)(119886))

to (1198642 (F2)(119886)) for each 119886 isin [0 1)

Theorem 36 Let (119864F) be a fuzzifying greedoid Then forall119886 isin(0 1] (119864F[119886]) is a greedoid Since 119864 is a finite set there is at

6 Journal of Function Spaces

most a finite number of greedoids that can be defined on119864Thusthere is a finite sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

such that

(1) if 119887 119888 isin (119886119894minus1 119886119894] (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 le 119886

119894lt 119888 le 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]

The sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof For 119886 119887 isin (0 1] we can easily check that if 119886 le 119887 thenF[119886]supe F[119887] We define an equivalence relation sim on (0 1]

by 119886 sim 119887 hArr F[119886]= F[119887] Since 119864 is a finite set the number

of greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is

a left open and right closed interval It is trivial that each119868119894(119894 = 1 2 119899) is an interval Suppose that 119886

119894= sup 119868

119894

and (119864F119894) is the corresponding cut greedoid with respect

to 119868119894 Then F

119894supe F[119886119894]

holds For all 119860 isin F119894 119860 isin F

[119903]

and F(119860) ge 119903 for all 119903 isin 119868119894 It follows that F(119860) ge 119886

119894and

119860 isin F[119886119894]

Thus F119894sube F[119886119894]

Hence F119894= F[119886119894]

and 119886119894isin 119868119894

Obviously the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is

the fundamental sequence for (119864F)

5 FYG as a Subcategory of IG

In this section we will study the relation between fuzzifyinggreedoids and fuzzy greedoids from the viewpoint of categorytheory

Theorem 37 Let (119864F) be a fuzzifying greedoid and 120590(F) =119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then

(1) 120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) (119864 120590(F)) is a closed and perfect 119868-greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860[119886]isin 120590(F)[119886] where

119860 isin 120590(F) Then 119860[119886]isin F[119886]

by the definition of 120590(F) Thismeans that120590(F)[119886] sube F

[119886](forall119886 isin (0 1]) Conversely assume

that 119860 isin F[119886] It is obvious that (119886 and 119860)

[119887]= 119860 isin F

[119886]sube

F[119887]

for every 119886 ge 119887 and (119886 and 119860)[119887]= 0 isin F

[119887]for every

119886 lt 119887 Hence 119886 and 119860 isin 120590(F) So 119860 = (119886 and 119860)[119886]isin 120590(F)[119886]

This implies that 120590(F)[119886] supe F[119886](forall119886 isin (0 1]) Therefore

120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) It is easy to see that 120590(F) satisfies (IG1) Nowwe provethat 120590(F) satisfies (IG2) Suppose that 119860 119861 isin 120590(F) and|119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|[119887] We

know that |119861[119887]| gt |119860

[119887]| Since119860

[119887] 119861[119887]isin F[119886]

and (119864F[119887])

is a greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup 119890 isin

F[119887] In this case ((119887 and 119860

[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F

[119887] It is

obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F

[119887]sube F[119886]

for every 119886 le 119887 and ((119887 and 119860[119887]) cup 119890119887)[119886]= 0 isin F

[119886]for every

119886 gt 119887 This implies that (119887 and 119860[119887]) cup 119890119887isin 120590(F)

By (1) and the definition of 120590(F) 120590(F) = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

120590(F)[119886] By Corollary 22 (119864 120590(F)) is a perfect 119868-greedoidByTheorem 36 (119864 120590(F)) is a closed 119868-greedoid

Theorem 38 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an 119868-feasibility preserving

mapping from (1198641 120590(F1)) to (119864

2 120590(F2))

Proof Since 119891 is a fuzzifying feasibility preserving mappingfrom (119864

1F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

by Corollary 35 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120590(F1)[119886]) to (119864

2 120590(F2)[119886]) for each

119886 isin (0 1] Since (1198641 120590(F1)) is perfect by Theorem 24 119891

is an 119868-feasibility preserving mapping from (1198641 120590(F1)) to

(1198642 120590(F2))

Theorem 39 Let (119864F) be a closed and perfect 119868-greedoidDefine a map 120575(F) 2119864 rarr 119868 by

120575 (F) (119860) = ⋁119886 isin (0 1] 119860 isinF [119886] (11)

Then

(1) 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) (119864 120575(F)) is a fuzzifying greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860 isin F[119886] Then120575(F)(119860) ge 119886 by the definition of 120575(F) We have119860 isin 120575(F)

[119886]

This means that 120575(F)[119886]supe F[119886] Conversely assume that

119860 isin 120575(F)[119886] that is 120575(F)(119860) ge 119886 Since (119864F) is a closed 119868-

greedoid there exists 119887 isin (0 1] such that 119887 ge 119886 and119860 isin F[119887]By 119887 ge 119886 F[119886] supe F[119887] This means that 120575(F)

[119886]sube F[119886]

Therefore 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) By (1) and Corollary 31 (119864 120575(F)) is a fuzzifyinggreedoid

Theorem 40 Let (119864119894F119894) (119894 = 1 2) be closed and perfect 119868-

greedoids If 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) then119891 is a fuzzifying feasibility

preserving mapping from (1198641 120575(F1)) to (119864

2 120575(F2))

Proof Since 119891 is an 119868-feasibility preserving mapping from(1198641F1) to (119864

2F2) 119891 is a feasibility preserving mapping

from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1] by

Theorem 24 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120575(F1)[119886]) to (119864

2 120575(F2)[119886]) for each 119886 isin

(0 1] By Corollary 35 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120575(F1)) to (119864

2 120575(F2))

Theorem 41 (1) If (119864F) is a fuzzifying greedoid then 120575 ∘120590(F) = F

(2) If (119864F) is a closed and perfect 119868-greedoid then 120590 ∘120575(F) = F

Proof (1) ByTheorem 37 (119864 120590(F)) is a closed and perfect 119868-greedoidThen 120575∘120590(F)

[119886]= 120590(F)[119886] = F

[119886]for all 119886 isin (0 1]

byTheorems 37 and 39 Hence 120575 ∘ 120590(F) = F(2) By Theorems 37 and 39 (119864 120590 ∘ 120575(F)) is a closed and

perfect 119868-greedoid Then 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1]119860[119886]

isin 120575(F)[119886] Since (119864F) is a closed and perfect 119868-

greedoid 120575(F)[119886]= F[119886] for all 119886 isin (0 1] by Theorem 39

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

Definition 7 (see [19]) A subfamilyI of 119871119864 is called a familyof independent 119871-fuzzy sets on 119864 if it satisfies the followingconditions

(LI1) 1205940isin I

(LI2) 119860 isin 119871119864 119861 isin I and 119860 sube 119861 rArr 119860 isin I(LI3) If 119860 119861 isin I and 119887 = |119861|(119899) 997408 |119860|(119899) for some 119899 isin N

then there exists 119890 isin 119865(119860 119861) such that (119887and119860[119887])cup119890119887isin

I where 119865(119860 119861) = 119890 isin 119864 119887 ⩽ 119861(119890) 119887 997408 119860(119890)

If I is a family of independent 119871-fuzzy sets on 119864 then thepair (119864I) is called an 119871-matroid

Definition 8 (see [19]) A mappingI 119871119864

rarr 119872 is called an119872-fuzzy family of independent 119871-fuzzy sets on 119864 if it satisfiesthe following conditions

(LMFI1)I(1205940) = ⊤119872

(LMFI2) For any 119860 119861 isin 119871119864 119860 sube 119861 rArr I(119860) ⩾ I(119861)(LMFI3) If 119887 = |119861|(119899) 997408 |119860|(119899) for 119860 119861 isin 119871119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

I ((119887 and 119860[119887]) cup 119890119887) ⩾ I (119860) andI (119861) (7)

IfI is an119872-fuzzy family of independent 119871-fuzzy sets on 119864then the pair (119864I) is called an (119871119872)-fuzzy matroid

Remark 9 (1) In [13 19]119871 and119872 denote completely distribu-tive lattices In [22] when 119871 = 119872 = [0 1] an (119871119872)-fuzzymatroid is also called 119887119894-fuzzy prematroid an 119871-matroidis called [0 1]-prematroid and an 119872-fuzzifying matroid iscalled a fuzzifying matroid

(2) When 119871 is replaced by the interval [0 1] it is easy tosee 119869(119871) = 119871 perp

119871 = (0 1]

Definition 10 (see [19]) Let 119860 be an 119871-fuzzy set on a finiteset 119864 Then the mapping |119860| N rarr 119871 defined by forall119899 isin N|119860|(119899) = ⋁119886 isin 119871 |119860

[119886]| ge 119899 is called the 119871-fuzzy

cardinality of 119860

Lemma 11 (see [19]) For a finite set 119864 it holds that |119860|[119886]=

|119860[119886]| for any 119860 isin 119871119864 and any 119886 isin 119869(119871)

Lemma 12 (see [19]) Let119866 sube 119864Then |119887and119866|(119899) = (119887and|119866|)(119899)for any 119899 gt 0 and for any 119887 isin 119871 perp

119871

3 119868-Greedoids

Based onDefinitions 2 and 7 we give the following definition

Definition 13 AsubfamilyF of 119868119864 is called a family of feasiblefuzzy sets on 119864 if it satisfies the following conditions

(IG1) 1205940isin F

(IG2) If 119860 119861 isin F and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin Nthen there exists 119890 isin 119865(119860 119861) such that (119887and119860

[119887])cup119890119887isin

F where

119865 (119860 119861) = 119890 isin 119864 119861 (119890) ge 119887 gt 119860 (119890) (8)

IfF is a family of feasible fuzzy sets on119864 then the pair (119864F)is called an 119868-greedoid

Theorem 14 Let F sube 119868119864 forall119886 isin (0 1] define F[119886] = 119860

[119886]

119860 isin F If (119864F) is an 119868-greedoid then (119864F[119886]) is a greedoidfor any 119886 isin (0 1]

Proof Suppose that (119864F) is an 119868-greedoid Now we provethat forall119886 isin (0 1] (119864F[119886]) is a greedoid

(1) Obviously 0 = (1205940)[119886]isin F[119886] for any 119886 isin (0 1]

(2) If119860 119861 isin F[119886] and |119860| lt |119861| then there are119866119867 isin Fsuch that119860 = 119866

[119886] 119861 = 119867

[119886] and |119866

[119886]| lt |119867

[119886]| Next

we prove 119886 and 119866[119886] 119886 and 119867

[119886]isin F

If 119866[119886]= 0 then it is obviously 119886 and 119866

[119886]isin F

If |119866[119886]| = 119899 ge 1 let119866

[119886]= 1198901 1198902 119890119899 Since 120594

0 119866 isin F

it is easy to see that there exists 1198991isin N such that |119866|(119899

1) ge

119886 gt |1205940|(1198991) By (IG2) there exists 119890

1isin 119865(120594

0 119866) such that

(119886 and (1205940)[119886]) cup (1198901)119886isin F where 119865(120594

0 119866) = 119890 isin 119864 119866(119890) ge

119886 gt 1205940(119890) We get (119890

1)119886isin F and 119890

1isin 119866[119886]

Since (1198901)119886 119866 isin F it is easy to see that there exists 119899

2isin N

such that |119866|(1198992) ge 119886 gt |(119890

1)119886|(1198992) By (IG2) there exists

1198902isin 119865((119890

1)119886 119866) such that (119886 and ((119890

1)119886)[119886]) cup (1198902)119886isin F where

119865((1198901)119886 119866) = 119890 isin 119864 119866(119890) ge 119886 gt (119890

1)119886(119890) We get 119886 and

1198901 1198902 isin F and 119890

1 1198902 sube 119866[119886]

Continue the above process and we can obtain 119886 and 119866[119886]isin

FAnalogously we can obtain 119886 and 119867

[119886]isin F

Since |119866[119886]| lt |119867

[119886]| take 119899 isin N such that |119866

[119886]| lt 119899 le

|119867[119886]|Then by Lemma 12 |119886and119866

[119886]|(119899) = (119886and|119866

[119886]|)(119899) = 0 lt

119886 = (119886and|119867[119886]|)(119899) = |119886and119867

[119886]|(119899) Since 119886and119866

[119886] 119886and119867

[119886]isin F

there is 119890 isin 119864 such that (119886 and 119867[119886])(119890) gt (119886 and 119866

[119886])(119890) and

(119886 and (119886 and 119866[119886])[119886]) cup 119890119886isin F It is easy to see that 119866

[119886](119890) = 0

119867[119886](119890) = 1 and (119886 and 119866

[119886]) cup 119890119886= (119886 and (119886 and 119866

[119886])[119886]) cup 119890119886isin F

This implies that 119890 isin 119867[119886]= 119861 119890 notin 119866

[119886]= 119860 and 119860 cup 119890 =

119866[119886]cup 119890 = ((119886 and 119866

[119886])[119886]cup 119890119886)[119886]isin F[119886]

Therefore (119864F[119886]) is a greedoid for any 119886 isin (0 1]

Corollary 15 Let (119864F) be an 119868-greedoid If 119860 isin F thenforall119886 isin (0 1] 119886 and 119860

[119886]isin F

Remark 16 If (119864F) is an [0 1]-prematroid it is obvious tosee that Corollary 15 holds by (LI2) An 119868-greedoid (119864F)need not satisfy (LI2) However by Theorem 14 Corollary 15still holds This is an important property for 119868-greedoidsMany results of 119868-greedoids in this paper are based on thisproperty

Definition 17 An 119868-greedoid (119864F) is called a perfect 119868-greedoid if it satisfies the following condition forall119860 isin 119868

119864 ifforall119886 isin (0 1] 119886 and 119860

[119886]isin F then 119860 isin F

ByCorollary 15 andDefinition 17 we can easily obtain thefollowing corollary

Corollary 18 An 119868-greedoid (119864F) is a perfect 119868-greedoid ifit satisfies the following condition forall119860 isin 119868

119864 if forall119886 isin (0 1]119860[119886]isin F[119886] then 119860 isin F

4 Journal of Function Spaces

Example 19 Let 119864 = 3 4 5 Define 119860 isin 119868119864 by

119860 (119890) =

0 119890 = 3

1

3 119890 = 4

1

2 119890 = 5

(9)

andF sube 119868119864 byF = 119861 isin 119868

119864

119861 le 12 and 5 cup 119861 isin 119868119864

119861 le

13 and 3 4 5 minus 13 and 3 It is easy to verify that (119864F) isan 119868-greedoid but it is not perfect since 119886 and 119860

[119886]isin F for all

119886 isin (0 1] but 119860 notin F

Lemma 20 Let (119864F) be an 119868-greedoid If 0 lt 119886 le 119887 le 1thenF[119886] supe F[119887]

Proof Let 119860 isin F[119887] By the definition of F[119887] there exists119866 isin F such that 119860 = 119866

[119887] By Corollary 15 we have 119887 and 119860 =

119887 and 119866[119887]isin F Then 119860 = (119887 and 119860)

[119886]isin F[119886]

Theorem 21 Let (119864F) be an 119868-greedoid DefineFlowast = 119860 isin119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then (119864Flowast) is an 119868-greedoid

Moreover if (119864F) is perfect thenF = Flowast

Proof (1) Obviously (1205940)[119886]= 0 isin F[119886] for any 119886 isin (0 1]

Thus 1205940isin Flowast

(2) Let 119860 119861 isin Flowast and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isinN Then |119861|

[119887]gt |119860|[119887] By Lemma 11 we know that |119861

[119887]| gt

|119860[119887]| Since 119860

[119887] 119861[119887]isin F[119887] and (119864F[119887]) is a greedoid

there exists 119890 isin 119861[119887]minus 119860[119887]

such that 119860[119887]cup 119890 isin F[119887] In

this case ((119887 and119860[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F[119887] It is obvious

that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F[119887] sube F[119886] for every

119886 le 119887 and ((119887 and119860[119887]) cup 119890119887)[119886]= 0 isin F[119886] for every 119886 gt 119887 This

implies that (119887 and 119860[119887]) cup 119890119887isin Flowast

(3) It is obvious thatF supe Flowast Since (119864F) is perfect byCorollary 18F sube Flowast HenceF = Flowast

Corollary 22 Let (119864F) be an 119868-greedoid Then (119864F) isperfect if and only ifF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886]

Definition 23 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids A map-

ping 119891 1198641rarr 1198642is called an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) if119891larr119868(119860) isin F

1for all119860 isin F

2

where 119891larr119868(119860) = 119860 ∘ 119891

Theorem 24 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids and let

119891 1198641rarr 119864

2be an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) If (119864

1F1) is a perfect 119868-greedoid then

the following conditions are equivalent

(1) 119891 is an 119868-feasibility preserving mapping from (1198641F1)

to (1198642F2)

(2) 119891 is a feasibility preserving mapping from (1198641F1[119886])

to (1198642F2[119886]) for each 119886 isin (0 1]

Proof (1) rArr (2) Let 119860 isin F2[119886] Then there exists 119860 isin F

2

such that 119860 = 119860[119886]isin F2[119886] Since 119891 is an 119868-feasibility pre-

servingmapping from (1198641F1) to (119864

2F2)119891larr119868(119860) isin F

1We

have 119891larr119868(119860)[119886]= 119891minus1

(119860[119886]) = 119891

minus1

(119860) isin F1[119886] This implies

that 119891 is a feasibility preserving mapping from (1198641F1[119886]) to

(1198642F2[119886]) for each 119886 isin (0 1]

(1) lArr (2) Assume that 119891 is a feasibility preservingmapping from (119864

1F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

Let 119860 isin F2 Then 119860

[119886]isin F2[119886] for all 119886 isin (0 1] Thus

119891larr

119868(119860)[119886]= 119891minus1

(119860[119886]) isin F

1[119886] for all 119886 isin (0 1] Since

(1198641F1) is a perfect 119868-greedoid by Corollary 18 119891larr

119868(119860) isin

F1 Therefore 119891 is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2)

Theorem 25 Let (119864F) be an 119868-greedoid Then forall119886 isin (0 1](119864F[119886]) is a greedoid Since 119864 is a set finite there is at most afinite number of greedoids on 119864 Thus there is a finite sequence0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 such that

(1) if 119887 119888 isin (119886119894minus1 119886119894) (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 lt 119886

119894lt 119888 lt 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]The sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof We define an equivalence relation sim on (0 1] by 119886 sim119887 hArr F[119886] = F[119887] Since 119864 is a finite set the numberof greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is an

interval We only need to show that forall119886 119887 isin 119868119894with 119886 le 119887 if

119888 isin [119886 119887] then 119888 isin 119868119894 Since 119886 le 119888 le 119887 by Lemma 20 we know

that F[119886] supe F[119888] supe F[119887] As 119886 119887 isin 119868119894 F[119886] = F[119888] Thus

119888 isin 119868119894by the definition of 119868

119894 This implies that 119868

119894is an interval

Let 119886119894minus1= inf 119868

119894and 119886

119894= sup 119868

119894(119894 = 1 2 119899) Obviously

the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is the

fundamental sequence for (119864F)

Definition 26 An 119868-greedoid (119864F) with the fundamentalsequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called a

closed 119868-greedoid if whenever 119886119894minus1lt 119886 le 119886

119894(1 le 119894 le 119903) then

F[119886] = F[119886119894]

Theorem 27 Let (119864F) be an 119868-greedoid with the fundamen-tal sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 Then (119864F)

is a closed 119868-greedoid if and only if it satisfies the followingcondition

(C) forall119886 isin (0 1] and 119860 isin 2119864 if 119887 and 119860 isin F for all

0 lt 119887 lt 119886 then 119886 and 119860 isin F

Proof Suppose that (119864F) satisfies (C) forall119886 isin (119886119894minus1 119886119894) (119894 =

1 2 119899) then F[119886] supe F[119886119894] Let 119860 isin F[119886] for all 119886 isin

(119886119894minus1 119886119894) Then 119886and119860 isin F for all 119886 isin (119886

119894minus1 119886119894) thus 119887 and119860 isin F

for all 0 lt 119887 lt 119886119894 SinceF satisfies (C) 119886

119894and 119860 isin F Thus 119860 =

(119886119894and119860)[119886119894]isin F[119886

119894]This implies thatF[119886] sube F[119886

119894] for all 119886 isin

(119886119894minus1 119886119894) Therefore F[119886] = F[119886

119894] for all 119886 isin (119886

119894minus1 119886119894] that

is (119864F) is closed Conversely assume that (119864F) is a closed119868-greedoid Let 119886 isin (0 1] 119860 isin 2119864 and 119887 and 119860 isin F for all 0 lt119887 lt 119886 Since 119886 isin (0 1] 119886 isin (119886

119894minus1 119886119894] for some 119894 = 1 2 119899

Take 1198870isin (119886119894minus1 119886) sube (119886

119894minus1 119886119894] Then 119887

0and 119860 isin F and thus

119860 isin F[1198870] Since (119864F) is closed 119860 isin F[119886

119894] Hence 119886

119894and119860 isin

F By Corollary 15 we have 119886 and119860 = 119886 and (119886119894and119860)[119886]isin F This

means thatF satisfies (C)

Journal of Function Spaces 5

4 (119868 119868)-Fuzzy Greedoids andFuzzifying Greedoids

Definition 28 AmappingF 119868119864

rarr 119868 is called a fuzzy familyof feasible fuzzy sets on 119864 if it satisfies the following condi-tions

(IIFG1)F(1205940) = 1

(IIFG2) If |119861|(119899) ge 119887 gt |119860|(119899) for 119860 119861 isin 119868119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

F ((119887 and 119860[119887]) cup 119890119887) ⩾ F (119860) andF (119861) (10)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890)

IfF is a fuzzy family of feasible fuzzy sets on 119864 then the pair(119864F) is called an (119868 119868)-fuzzy greedoid

Definition 29 AmappingF 2119864

rarr 119868 is called a fuzzy familyof feasible sets on 119864 if it satisfies the following conditions

(FYG1)F(0) = 1

(FYG2) For any 119860 119861 isin 2119864 if |119860| lt |119861| then

⋁119890isin119861minus119860

F(119860 cup 119890) ⩾ F(119860) andF(119861)

IfF is a fuzzy family of feasible sets on119864 then the pair (119864F)is called a fuzzifying greedoid For 119860 isin 2

119864 F(119860) can beregarded as the degree to which 119860 is a feasible set

Theorem 30 Let F 119868119864

rarr 119868 be a mapping Then the fol-lowing conditions are equivalent

(1) (119864F) is an (119868 119868)-fuzzy greedoid

(2) For each 119886 isin (0 1] (119864F[119886]) is an 119868-greedoid

(3) For each 119886 isin [0 1) (119864F(119886)) is an 119868-greedoid

Proof (1) rArr (2) (IG1) It is obvious that 1205940isin F[119886]

for any119886 isin (0 1]

(IG2) If 119860 119861 isin F[119886]

and |119861|(119899) ge 119887 gt |119860|(119899) for some119899 isin N then by (IIFG2) we have ⋁

119890isin119865(119860119861)F((119887 and 119860

[119887]) cup

119890119887) ⩾ F(119860) and F(119861) Further by 119860 119861 isin F

[119886] we obtain

⋁119890isin119865(119860119861)

F((119887 and119860[119887]) cup 119890119887) ⩾ 119886 Then there exists 119890 isin 119865(119860 119861)

such thatF((119887and119860[119887])cup119890119887) ⩾ 119886 that is (119887and119860

[119887])cup119890119887isin F[119886]

This shows thatF[119886]

satisfies (IG2) Therefore (119864F[119886]) is an

119868-greedoid for each 119886 isin (0 1](2) rArr (1) (IIFG1) For any 119886 isin (0 1] 120594

0isin F[119886] We have

F(1205940) = 1

(IIFG2) Suppose that 119860 119861 isin [0 1]119864 and |119861|(119899) ge 119887 gt|119860|(119899) for some 119899 isin N In order to prove (IIFG2) take 119886 isin(0 1] andF(119860) andF(119861) ge 119886 ThenF(119860) ge 119886 andF(119861) ge 119886This implies 119860 119861 isin F

[119886] Hence by (IG2) there exists 119890 isin

119865(119860 119861) such that (119887 and 119860[119887]) cup 119890119887isin F[119886] This impliesF((119887 and

119860[119887])cup119890119887) ⩾ 119886 Furtherwehave⋁

119890isin119865(119860119861)F((119887and119860

[119887])cup119890119887) ⩾ 119886

By the arbitrariness of 119886 we obtain or119890isin119865(119860119861)

F((119887 and 119860[119887]) cup

119890119887) ⩾ F(119860) andF(119861)Analogously we can obtain (1) hArr (3)

Corollary 31 Let F 2119864

rarr 119868 be a map Then the followingconditions are equivalent

(1) (119864F) is a fuzzifying greedoid(2) (119864F

[119886]) is a greedoid for each 119886 isin (0 1]

(3) (119864F(119886)) is a greedoid for each 119886 isin [0 1)

Definition 32 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

A mapping 119891 1198641

rarr 1198642is called an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2) if

F1(119891larr

119868(119860)) ge F

2(119860) for all 119860 isin 1198681198642 where 119891larr

119868(119860) = 119860 ∘ 119891

Definition 33 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

A mapping 119891 1198641rarr 1198642is called a fuzzifying feasibility pre-

serving mapping from (1198641F1) to (119864

2F2) ifF

1(119891minus1

(119860)) ge

F2(119860) for all 119860 isin 21198642

Theorem 34 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

and let 119891 1198641rarr 1198642be an (119868 119868)-fuzzy feasibility preserving

mapping from (1198641F1) to (119864

2F2) Then the following condi-

tions are equivalent

(1) 119891 is an (119868 119868)-fuzzy feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)(119886)) to (119864

2 (F2)(119886)) for each 119886 isin [0 1)

Proof (1) rArr (2) Let119891 be an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1F1) to (119864

2F2) and 119886 isin (0 1] forall119860 isin

(F2)[119886] we have F

1(119891larr

119868(119860)) ge F

2(119860) ge 119886 and 119891larr

119868(119860) isin

(F1)[119886] In other words 119891 is a fuzzy feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin

(0 1](1) rArr (2) Next we need to prove that F

1(119891larr

119868(119860)) ge

F2(119860) for any 119860 isin 119868

119864 Let F2(119860) ge 119886 Then 119860 isin

(F2)[119886] Since 119891 is an 119868-feasibility preserving mapping from

(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1] 119891larr

119868(119860) isin

(F1)[119886] that is F

1(119891larr

119868(119860)) ge 119886 By the arbitrariness of 119886

we haveF1(119891larr

119868(119860)) ge F

2(119860) Therefore 119891 is an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2)

Analogously we can obtain (1) hArr (3)

Corollary 35 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

and let119891 1198641rarr 1198642be a fuzzifying feasibility preservingmap-

ping from (1198641F1) to (119864

2F2) Then the following conditions

are equivalent

(1) 119891 is a fuzzifying feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is a feasibility preserving mapping from (1198641 (F1)[119886])

to (1198642 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is a feasibility preserving mapping from (1198641 (F1)(119886))

to (1198642 (F2)(119886)) for each 119886 isin [0 1)

Theorem 36 Let (119864F) be a fuzzifying greedoid Then forall119886 isin(0 1] (119864F[119886]) is a greedoid Since 119864 is a finite set there is at

6 Journal of Function Spaces

most a finite number of greedoids that can be defined on119864Thusthere is a finite sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

such that

(1) if 119887 119888 isin (119886119894minus1 119886119894] (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 le 119886

119894lt 119888 le 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]

The sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof For 119886 119887 isin (0 1] we can easily check that if 119886 le 119887 thenF[119886]supe F[119887] We define an equivalence relation sim on (0 1]

by 119886 sim 119887 hArr F[119886]= F[119887] Since 119864 is a finite set the number

of greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is

a left open and right closed interval It is trivial that each119868119894(119894 = 1 2 119899) is an interval Suppose that 119886

119894= sup 119868

119894

and (119864F119894) is the corresponding cut greedoid with respect

to 119868119894 Then F

119894supe F[119886119894]

holds For all 119860 isin F119894 119860 isin F

[119903]

and F(119860) ge 119903 for all 119903 isin 119868119894 It follows that F(119860) ge 119886

119894and

119860 isin F[119886119894]

Thus F119894sube F[119886119894]

Hence F119894= F[119886119894]

and 119886119894isin 119868119894

Obviously the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is

the fundamental sequence for (119864F)

5 FYG as a Subcategory of IG

In this section we will study the relation between fuzzifyinggreedoids and fuzzy greedoids from the viewpoint of categorytheory

Theorem 37 Let (119864F) be a fuzzifying greedoid and 120590(F) =119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then

(1) 120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) (119864 120590(F)) is a closed and perfect 119868-greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860[119886]isin 120590(F)[119886] where

119860 isin 120590(F) Then 119860[119886]isin F[119886]

by the definition of 120590(F) Thismeans that120590(F)[119886] sube F

[119886](forall119886 isin (0 1]) Conversely assume

that 119860 isin F[119886] It is obvious that (119886 and 119860)

[119887]= 119860 isin F

[119886]sube

F[119887]

for every 119886 ge 119887 and (119886 and 119860)[119887]= 0 isin F

[119887]for every

119886 lt 119887 Hence 119886 and 119860 isin 120590(F) So 119860 = (119886 and 119860)[119886]isin 120590(F)[119886]

This implies that 120590(F)[119886] supe F[119886](forall119886 isin (0 1]) Therefore

120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) It is easy to see that 120590(F) satisfies (IG1) Nowwe provethat 120590(F) satisfies (IG2) Suppose that 119860 119861 isin 120590(F) and|119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|[119887] We

know that |119861[119887]| gt |119860

[119887]| Since119860

[119887] 119861[119887]isin F[119886]

and (119864F[119887])

is a greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup 119890 isin

F[119887] In this case ((119887 and 119860

[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F

[119887] It is

obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F

[119887]sube F[119886]

for every 119886 le 119887 and ((119887 and 119860[119887]) cup 119890119887)[119886]= 0 isin F

[119886]for every

119886 gt 119887 This implies that (119887 and 119860[119887]) cup 119890119887isin 120590(F)

By (1) and the definition of 120590(F) 120590(F) = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

120590(F)[119886] By Corollary 22 (119864 120590(F)) is a perfect 119868-greedoidByTheorem 36 (119864 120590(F)) is a closed 119868-greedoid

Theorem 38 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an 119868-feasibility preserving

mapping from (1198641 120590(F1)) to (119864

2 120590(F2))

Proof Since 119891 is a fuzzifying feasibility preserving mappingfrom (119864

1F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

by Corollary 35 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120590(F1)[119886]) to (119864

2 120590(F2)[119886]) for each

119886 isin (0 1] Since (1198641 120590(F1)) is perfect by Theorem 24 119891

is an 119868-feasibility preserving mapping from (1198641 120590(F1)) to

(1198642 120590(F2))

Theorem 39 Let (119864F) be a closed and perfect 119868-greedoidDefine a map 120575(F) 2119864 rarr 119868 by

120575 (F) (119860) = ⋁119886 isin (0 1] 119860 isinF [119886] (11)

Then

(1) 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) (119864 120575(F)) is a fuzzifying greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860 isin F[119886] Then120575(F)(119860) ge 119886 by the definition of 120575(F) We have119860 isin 120575(F)

[119886]

This means that 120575(F)[119886]supe F[119886] Conversely assume that

119860 isin 120575(F)[119886] that is 120575(F)(119860) ge 119886 Since (119864F) is a closed 119868-

greedoid there exists 119887 isin (0 1] such that 119887 ge 119886 and119860 isin F[119887]By 119887 ge 119886 F[119886] supe F[119887] This means that 120575(F)

[119886]sube F[119886]

Therefore 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) By (1) and Corollary 31 (119864 120575(F)) is a fuzzifyinggreedoid

Theorem 40 Let (119864119894F119894) (119894 = 1 2) be closed and perfect 119868-

greedoids If 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) then119891 is a fuzzifying feasibility

preserving mapping from (1198641 120575(F1)) to (119864

2 120575(F2))

Proof Since 119891 is an 119868-feasibility preserving mapping from(1198641F1) to (119864

2F2) 119891 is a feasibility preserving mapping

from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1] by

Theorem 24 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120575(F1)[119886]) to (119864

2 120575(F2)[119886]) for each 119886 isin

(0 1] By Corollary 35 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120575(F1)) to (119864

2 120575(F2))

Theorem 41 (1) If (119864F) is a fuzzifying greedoid then 120575 ∘120590(F) = F

(2) If (119864F) is a closed and perfect 119868-greedoid then 120590 ∘120575(F) = F

Proof (1) ByTheorem 37 (119864 120590(F)) is a closed and perfect 119868-greedoidThen 120575∘120590(F)

[119886]= 120590(F)[119886] = F

[119886]for all 119886 isin (0 1]

byTheorems 37 and 39 Hence 120575 ∘ 120590(F) = F(2) By Theorems 37 and 39 (119864 120590 ∘ 120575(F)) is a closed and

perfect 119868-greedoid Then 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1]119860[119886]

isin 120575(F)[119886] Since (119864F) is a closed and perfect 119868-

greedoid 120575(F)[119886]= F[119886] for all 119886 isin (0 1] by Theorem 39

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

Example 19 Let 119864 = 3 4 5 Define 119860 isin 119868119864 by

119860 (119890) =

0 119890 = 3

1

3 119890 = 4

1

2 119890 = 5

(9)

andF sube 119868119864 byF = 119861 isin 119868

119864

119861 le 12 and 5 cup 119861 isin 119868119864

119861 le

13 and 3 4 5 minus 13 and 3 It is easy to verify that (119864F) isan 119868-greedoid but it is not perfect since 119886 and 119860

[119886]isin F for all

119886 isin (0 1] but 119860 notin F

Lemma 20 Let (119864F) be an 119868-greedoid If 0 lt 119886 le 119887 le 1thenF[119886] supe F[119887]

Proof Let 119860 isin F[119887] By the definition of F[119887] there exists119866 isin F such that 119860 = 119866

[119887] By Corollary 15 we have 119887 and 119860 =

119887 and 119866[119887]isin F Then 119860 = (119887 and 119860)

[119886]isin F[119886]

Theorem 21 Let (119864F) be an 119868-greedoid DefineFlowast = 119860 isin119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then (119864Flowast) is an 119868-greedoid

Moreover if (119864F) is perfect thenF = Flowast

Proof (1) Obviously (1205940)[119886]= 0 isin F[119886] for any 119886 isin (0 1]

Thus 1205940isin Flowast

(2) Let 119860 119861 isin Flowast and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isinN Then |119861|

[119887]gt |119860|[119887] By Lemma 11 we know that |119861

[119887]| gt

|119860[119887]| Since 119860

[119887] 119861[119887]isin F[119887] and (119864F[119887]) is a greedoid

there exists 119890 isin 119861[119887]minus 119860[119887]

such that 119860[119887]cup 119890 isin F[119887] In

this case ((119887 and119860[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F[119887] It is obvious

that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F[119887] sube F[119886] for every

119886 le 119887 and ((119887 and119860[119887]) cup 119890119887)[119886]= 0 isin F[119886] for every 119886 gt 119887 This

implies that (119887 and 119860[119887]) cup 119890119887isin Flowast

(3) It is obvious thatF supe Flowast Since (119864F) is perfect byCorollary 18F sube Flowast HenceF = Flowast

Corollary 22 Let (119864F) be an 119868-greedoid Then (119864F) isperfect if and only ifF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886]

Definition 23 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids A map-

ping 119891 1198641rarr 1198642is called an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) if119891larr119868(119860) isin F

1for all119860 isin F

2

where 119891larr119868(119860) = 119860 ∘ 119891

Theorem 24 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids and let

119891 1198641rarr 119864

2be an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) If (119864

1F1) is a perfect 119868-greedoid then

the following conditions are equivalent

(1) 119891 is an 119868-feasibility preserving mapping from (1198641F1)

to (1198642F2)

(2) 119891 is a feasibility preserving mapping from (1198641F1[119886])

to (1198642F2[119886]) for each 119886 isin (0 1]

Proof (1) rArr (2) Let 119860 isin F2[119886] Then there exists 119860 isin F

2

such that 119860 = 119860[119886]isin F2[119886] Since 119891 is an 119868-feasibility pre-

servingmapping from (1198641F1) to (119864

2F2)119891larr119868(119860) isin F

1We

have 119891larr119868(119860)[119886]= 119891minus1

(119860[119886]) = 119891

minus1

(119860) isin F1[119886] This implies

that 119891 is a feasibility preserving mapping from (1198641F1[119886]) to

(1198642F2[119886]) for each 119886 isin (0 1]

(1) lArr (2) Assume that 119891 is a feasibility preservingmapping from (119864

1F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

Let 119860 isin F2 Then 119860

[119886]isin F2[119886] for all 119886 isin (0 1] Thus

119891larr

119868(119860)[119886]= 119891minus1

(119860[119886]) isin F

1[119886] for all 119886 isin (0 1] Since

(1198641F1) is a perfect 119868-greedoid by Corollary 18 119891larr

119868(119860) isin

F1 Therefore 119891 is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2)

Theorem 25 Let (119864F) be an 119868-greedoid Then forall119886 isin (0 1](119864F[119886]) is a greedoid Since 119864 is a set finite there is at most afinite number of greedoids on 119864 Thus there is a finite sequence0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 such that

(1) if 119887 119888 isin (119886119894minus1 119886119894) (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 lt 119886

119894lt 119888 lt 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]The sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof We define an equivalence relation sim on (0 1] by 119886 sim119887 hArr F[119886] = F[119887] Since 119864 is a finite set the numberof greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is an

interval We only need to show that forall119886 119887 isin 119868119894with 119886 le 119887 if

119888 isin [119886 119887] then 119888 isin 119868119894 Since 119886 le 119888 le 119887 by Lemma 20 we know

that F[119886] supe F[119888] supe F[119887] As 119886 119887 isin 119868119894 F[119886] = F[119888] Thus

119888 isin 119868119894by the definition of 119868

119894 This implies that 119868

119894is an interval

Let 119886119894minus1= inf 119868

119894and 119886

119894= sup 119868

119894(119894 = 1 2 119899) Obviously

the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is the

fundamental sequence for (119864F)

Definition 26 An 119868-greedoid (119864F) with the fundamentalsequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called a

closed 119868-greedoid if whenever 119886119894minus1lt 119886 le 119886

119894(1 le 119894 le 119903) then

F[119886] = F[119886119894]

Theorem 27 Let (119864F) be an 119868-greedoid with the fundamen-tal sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 Then (119864F)

is a closed 119868-greedoid if and only if it satisfies the followingcondition

(C) forall119886 isin (0 1] and 119860 isin 2119864 if 119887 and 119860 isin F for all

0 lt 119887 lt 119886 then 119886 and 119860 isin F

Proof Suppose that (119864F) satisfies (C) forall119886 isin (119886119894minus1 119886119894) (119894 =

1 2 119899) then F[119886] supe F[119886119894] Let 119860 isin F[119886] for all 119886 isin

(119886119894minus1 119886119894) Then 119886and119860 isin F for all 119886 isin (119886

119894minus1 119886119894) thus 119887 and119860 isin F

for all 0 lt 119887 lt 119886119894 SinceF satisfies (C) 119886

119894and 119860 isin F Thus 119860 =

(119886119894and119860)[119886119894]isin F[119886

119894]This implies thatF[119886] sube F[119886

119894] for all 119886 isin

(119886119894minus1 119886119894) Therefore F[119886] = F[119886

119894] for all 119886 isin (119886

119894minus1 119886119894] that

is (119864F) is closed Conversely assume that (119864F) is a closed119868-greedoid Let 119886 isin (0 1] 119860 isin 2119864 and 119887 and 119860 isin F for all 0 lt119887 lt 119886 Since 119886 isin (0 1] 119886 isin (119886

119894minus1 119886119894] for some 119894 = 1 2 119899

Take 1198870isin (119886119894minus1 119886) sube (119886

119894minus1 119886119894] Then 119887

0and 119860 isin F and thus

119860 isin F[1198870] Since (119864F) is closed 119860 isin F[119886

119894] Hence 119886

119894and119860 isin

F By Corollary 15 we have 119886 and119860 = 119886 and (119886119894and119860)[119886]isin F This

means thatF satisfies (C)

Journal of Function Spaces 5

4 (119868 119868)-Fuzzy Greedoids andFuzzifying Greedoids

Definition 28 AmappingF 119868119864

rarr 119868 is called a fuzzy familyof feasible fuzzy sets on 119864 if it satisfies the following condi-tions

(IIFG1)F(1205940) = 1

(IIFG2) If |119861|(119899) ge 119887 gt |119860|(119899) for 119860 119861 isin 119868119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

F ((119887 and 119860[119887]) cup 119890119887) ⩾ F (119860) andF (119861) (10)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890)

IfF is a fuzzy family of feasible fuzzy sets on 119864 then the pair(119864F) is called an (119868 119868)-fuzzy greedoid

Definition 29 AmappingF 2119864

rarr 119868 is called a fuzzy familyof feasible sets on 119864 if it satisfies the following conditions

(FYG1)F(0) = 1

(FYG2) For any 119860 119861 isin 2119864 if |119860| lt |119861| then

⋁119890isin119861minus119860

F(119860 cup 119890) ⩾ F(119860) andF(119861)

IfF is a fuzzy family of feasible sets on119864 then the pair (119864F)is called a fuzzifying greedoid For 119860 isin 2

119864 F(119860) can beregarded as the degree to which 119860 is a feasible set

Theorem 30 Let F 119868119864

rarr 119868 be a mapping Then the fol-lowing conditions are equivalent

(1) (119864F) is an (119868 119868)-fuzzy greedoid

(2) For each 119886 isin (0 1] (119864F[119886]) is an 119868-greedoid

(3) For each 119886 isin [0 1) (119864F(119886)) is an 119868-greedoid

Proof (1) rArr (2) (IG1) It is obvious that 1205940isin F[119886]

for any119886 isin (0 1]

(IG2) If 119860 119861 isin F[119886]

and |119861|(119899) ge 119887 gt |119860|(119899) for some119899 isin N then by (IIFG2) we have ⋁

119890isin119865(119860119861)F((119887 and 119860

[119887]) cup

119890119887) ⩾ F(119860) and F(119861) Further by 119860 119861 isin F

[119886] we obtain

⋁119890isin119865(119860119861)

F((119887 and119860[119887]) cup 119890119887) ⩾ 119886 Then there exists 119890 isin 119865(119860 119861)

such thatF((119887and119860[119887])cup119890119887) ⩾ 119886 that is (119887and119860

[119887])cup119890119887isin F[119886]

This shows thatF[119886]

satisfies (IG2) Therefore (119864F[119886]) is an

119868-greedoid for each 119886 isin (0 1](2) rArr (1) (IIFG1) For any 119886 isin (0 1] 120594

0isin F[119886] We have

F(1205940) = 1

(IIFG2) Suppose that 119860 119861 isin [0 1]119864 and |119861|(119899) ge 119887 gt|119860|(119899) for some 119899 isin N In order to prove (IIFG2) take 119886 isin(0 1] andF(119860) andF(119861) ge 119886 ThenF(119860) ge 119886 andF(119861) ge 119886This implies 119860 119861 isin F

[119886] Hence by (IG2) there exists 119890 isin

119865(119860 119861) such that (119887 and 119860[119887]) cup 119890119887isin F[119886] This impliesF((119887 and

119860[119887])cup119890119887) ⩾ 119886 Furtherwehave⋁

119890isin119865(119860119861)F((119887and119860

[119887])cup119890119887) ⩾ 119886

By the arbitrariness of 119886 we obtain or119890isin119865(119860119861)

F((119887 and 119860[119887]) cup

119890119887) ⩾ F(119860) andF(119861)Analogously we can obtain (1) hArr (3)

Corollary 31 Let F 2119864

rarr 119868 be a map Then the followingconditions are equivalent

(1) (119864F) is a fuzzifying greedoid(2) (119864F

[119886]) is a greedoid for each 119886 isin (0 1]

(3) (119864F(119886)) is a greedoid for each 119886 isin [0 1)

Definition 32 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

A mapping 119891 1198641

rarr 1198642is called an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2) if

F1(119891larr

119868(119860)) ge F

2(119860) for all 119860 isin 1198681198642 where 119891larr

119868(119860) = 119860 ∘ 119891

Definition 33 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

A mapping 119891 1198641rarr 1198642is called a fuzzifying feasibility pre-

serving mapping from (1198641F1) to (119864

2F2) ifF

1(119891minus1

(119860)) ge

F2(119860) for all 119860 isin 21198642

Theorem 34 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

and let 119891 1198641rarr 1198642be an (119868 119868)-fuzzy feasibility preserving

mapping from (1198641F1) to (119864

2F2) Then the following condi-

tions are equivalent

(1) 119891 is an (119868 119868)-fuzzy feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)(119886)) to (119864

2 (F2)(119886)) for each 119886 isin [0 1)

Proof (1) rArr (2) Let119891 be an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1F1) to (119864

2F2) and 119886 isin (0 1] forall119860 isin

(F2)[119886] we have F

1(119891larr

119868(119860)) ge F

2(119860) ge 119886 and 119891larr

119868(119860) isin

(F1)[119886] In other words 119891 is a fuzzy feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin

(0 1](1) rArr (2) Next we need to prove that F

1(119891larr

119868(119860)) ge

F2(119860) for any 119860 isin 119868

119864 Let F2(119860) ge 119886 Then 119860 isin

(F2)[119886] Since 119891 is an 119868-feasibility preserving mapping from

(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1] 119891larr

119868(119860) isin

(F1)[119886] that is F

1(119891larr

119868(119860)) ge 119886 By the arbitrariness of 119886

we haveF1(119891larr

119868(119860)) ge F

2(119860) Therefore 119891 is an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2)

Analogously we can obtain (1) hArr (3)

Corollary 35 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

and let119891 1198641rarr 1198642be a fuzzifying feasibility preservingmap-

ping from (1198641F1) to (119864

2F2) Then the following conditions

are equivalent

(1) 119891 is a fuzzifying feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is a feasibility preserving mapping from (1198641 (F1)[119886])

to (1198642 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is a feasibility preserving mapping from (1198641 (F1)(119886))

to (1198642 (F2)(119886)) for each 119886 isin [0 1)

Theorem 36 Let (119864F) be a fuzzifying greedoid Then forall119886 isin(0 1] (119864F[119886]) is a greedoid Since 119864 is a finite set there is at

6 Journal of Function Spaces

most a finite number of greedoids that can be defined on119864Thusthere is a finite sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

such that

(1) if 119887 119888 isin (119886119894minus1 119886119894] (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 le 119886

119894lt 119888 le 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]

The sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof For 119886 119887 isin (0 1] we can easily check that if 119886 le 119887 thenF[119886]supe F[119887] We define an equivalence relation sim on (0 1]

by 119886 sim 119887 hArr F[119886]= F[119887] Since 119864 is a finite set the number

of greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is

a left open and right closed interval It is trivial that each119868119894(119894 = 1 2 119899) is an interval Suppose that 119886

119894= sup 119868

119894

and (119864F119894) is the corresponding cut greedoid with respect

to 119868119894 Then F

119894supe F[119886119894]

holds For all 119860 isin F119894 119860 isin F

[119903]

and F(119860) ge 119903 for all 119903 isin 119868119894 It follows that F(119860) ge 119886

119894and

119860 isin F[119886119894]

Thus F119894sube F[119886119894]

Hence F119894= F[119886119894]

and 119886119894isin 119868119894

Obviously the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is

the fundamental sequence for (119864F)

5 FYG as a Subcategory of IG

In this section we will study the relation between fuzzifyinggreedoids and fuzzy greedoids from the viewpoint of categorytheory

Theorem 37 Let (119864F) be a fuzzifying greedoid and 120590(F) =119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then

(1) 120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) (119864 120590(F)) is a closed and perfect 119868-greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860[119886]isin 120590(F)[119886] where

119860 isin 120590(F) Then 119860[119886]isin F[119886]

by the definition of 120590(F) Thismeans that120590(F)[119886] sube F

[119886](forall119886 isin (0 1]) Conversely assume

that 119860 isin F[119886] It is obvious that (119886 and 119860)

[119887]= 119860 isin F

[119886]sube

F[119887]

for every 119886 ge 119887 and (119886 and 119860)[119887]= 0 isin F

[119887]for every

119886 lt 119887 Hence 119886 and 119860 isin 120590(F) So 119860 = (119886 and 119860)[119886]isin 120590(F)[119886]

This implies that 120590(F)[119886] supe F[119886](forall119886 isin (0 1]) Therefore

120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) It is easy to see that 120590(F) satisfies (IG1) Nowwe provethat 120590(F) satisfies (IG2) Suppose that 119860 119861 isin 120590(F) and|119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|[119887] We

know that |119861[119887]| gt |119860

[119887]| Since119860

[119887] 119861[119887]isin F[119886]

and (119864F[119887])

is a greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup 119890 isin

F[119887] In this case ((119887 and 119860

[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F

[119887] It is

obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F

[119887]sube F[119886]

for every 119886 le 119887 and ((119887 and 119860[119887]) cup 119890119887)[119886]= 0 isin F

[119886]for every

119886 gt 119887 This implies that (119887 and 119860[119887]) cup 119890119887isin 120590(F)

By (1) and the definition of 120590(F) 120590(F) = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

120590(F)[119886] By Corollary 22 (119864 120590(F)) is a perfect 119868-greedoidByTheorem 36 (119864 120590(F)) is a closed 119868-greedoid

Theorem 38 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an 119868-feasibility preserving

mapping from (1198641 120590(F1)) to (119864

2 120590(F2))

Proof Since 119891 is a fuzzifying feasibility preserving mappingfrom (119864

1F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

by Corollary 35 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120590(F1)[119886]) to (119864

2 120590(F2)[119886]) for each

119886 isin (0 1] Since (1198641 120590(F1)) is perfect by Theorem 24 119891

is an 119868-feasibility preserving mapping from (1198641 120590(F1)) to

(1198642 120590(F2))

Theorem 39 Let (119864F) be a closed and perfect 119868-greedoidDefine a map 120575(F) 2119864 rarr 119868 by

120575 (F) (119860) = ⋁119886 isin (0 1] 119860 isinF [119886] (11)

Then

(1) 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) (119864 120575(F)) is a fuzzifying greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860 isin F[119886] Then120575(F)(119860) ge 119886 by the definition of 120575(F) We have119860 isin 120575(F)

[119886]

This means that 120575(F)[119886]supe F[119886] Conversely assume that

119860 isin 120575(F)[119886] that is 120575(F)(119860) ge 119886 Since (119864F) is a closed 119868-

greedoid there exists 119887 isin (0 1] such that 119887 ge 119886 and119860 isin F[119887]By 119887 ge 119886 F[119886] supe F[119887] This means that 120575(F)

[119886]sube F[119886]

Therefore 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) By (1) and Corollary 31 (119864 120575(F)) is a fuzzifyinggreedoid

Theorem 40 Let (119864119894F119894) (119894 = 1 2) be closed and perfect 119868-

greedoids If 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) then119891 is a fuzzifying feasibility

preserving mapping from (1198641 120575(F1)) to (119864

2 120575(F2))

Proof Since 119891 is an 119868-feasibility preserving mapping from(1198641F1) to (119864

2F2) 119891 is a feasibility preserving mapping

from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1] by

Theorem 24 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120575(F1)[119886]) to (119864

2 120575(F2)[119886]) for each 119886 isin

(0 1] By Corollary 35 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120575(F1)) to (119864

2 120575(F2))

Theorem 41 (1) If (119864F) is a fuzzifying greedoid then 120575 ∘120590(F) = F

(2) If (119864F) is a closed and perfect 119868-greedoid then 120590 ∘120575(F) = F

Proof (1) ByTheorem 37 (119864 120590(F)) is a closed and perfect 119868-greedoidThen 120575∘120590(F)

[119886]= 120590(F)[119886] = F

[119886]for all 119886 isin (0 1]

byTheorems 37 and 39 Hence 120575 ∘ 120590(F) = F(2) By Theorems 37 and 39 (119864 120590 ∘ 120575(F)) is a closed and

perfect 119868-greedoid Then 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1]119860[119886]

isin 120575(F)[119886] Since (119864F) is a closed and perfect 119868-

greedoid 120575(F)[119886]= F[119886] for all 119886 isin (0 1] by Theorem 39

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

4 (119868 119868)-Fuzzy Greedoids andFuzzifying Greedoids

Definition 28 AmappingF 119868119864

rarr 119868 is called a fuzzy familyof feasible fuzzy sets on 119864 if it satisfies the following condi-tions

(IIFG1)F(1205940) = 1

(IIFG2) If |119861|(119899) ge 119887 gt |119860|(119899) for 119860 119861 isin 119868119864 and forsome 119899 isin N then

⋁

119890isin119865(119860119861)

F ((119887 and 119860[119887]) cup 119890119887) ⩾ F (119860) andF (119861) (10)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890)

IfF is a fuzzy family of feasible fuzzy sets on 119864 then the pair(119864F) is called an (119868 119868)-fuzzy greedoid

Definition 29 AmappingF 2119864

rarr 119868 is called a fuzzy familyof feasible sets on 119864 if it satisfies the following conditions

(FYG1)F(0) = 1

(FYG2) For any 119860 119861 isin 2119864 if |119860| lt |119861| then

⋁119890isin119861minus119860

F(119860 cup 119890) ⩾ F(119860) andF(119861)

IfF is a fuzzy family of feasible sets on119864 then the pair (119864F)is called a fuzzifying greedoid For 119860 isin 2

119864 F(119860) can beregarded as the degree to which 119860 is a feasible set

Theorem 30 Let F 119868119864

rarr 119868 be a mapping Then the fol-lowing conditions are equivalent

(1) (119864F) is an (119868 119868)-fuzzy greedoid

(2) For each 119886 isin (0 1] (119864F[119886]) is an 119868-greedoid

(3) For each 119886 isin [0 1) (119864F(119886)) is an 119868-greedoid

Proof (1) rArr (2) (IG1) It is obvious that 1205940isin F[119886]

for any119886 isin (0 1]

(IG2) If 119860 119861 isin F[119886]

and |119861|(119899) ge 119887 gt |119860|(119899) for some119899 isin N then by (IIFG2) we have ⋁

119890isin119865(119860119861)F((119887 and 119860

[119887]) cup

119890119887) ⩾ F(119860) and F(119861) Further by 119860 119861 isin F

[119886] we obtain

⋁119890isin119865(119860119861)

F((119887 and119860[119887]) cup 119890119887) ⩾ 119886 Then there exists 119890 isin 119865(119860 119861)

such thatF((119887and119860[119887])cup119890119887) ⩾ 119886 that is (119887and119860

[119887])cup119890119887isin F[119886]

This shows thatF[119886]

satisfies (IG2) Therefore (119864F[119886]) is an

119868-greedoid for each 119886 isin (0 1](2) rArr (1) (IIFG1) For any 119886 isin (0 1] 120594

0isin F[119886] We have

F(1205940) = 1

(IIFG2) Suppose that 119860 119861 isin [0 1]119864 and |119861|(119899) ge 119887 gt|119860|(119899) for some 119899 isin N In order to prove (IIFG2) take 119886 isin(0 1] andF(119860) andF(119861) ge 119886 ThenF(119860) ge 119886 andF(119861) ge 119886This implies 119860 119861 isin F

[119886] Hence by (IG2) there exists 119890 isin

119865(119860 119861) such that (119887 and 119860[119887]) cup 119890119887isin F[119886] This impliesF((119887 and

119860[119887])cup119890119887) ⩾ 119886 Furtherwehave⋁

119890isin119865(119860119861)F((119887and119860

[119887])cup119890119887) ⩾ 119886

By the arbitrariness of 119886 we obtain or119890isin119865(119860119861)

F((119887 and 119860[119887]) cup

119890119887) ⩾ F(119860) andF(119861)Analogously we can obtain (1) hArr (3)

Corollary 31 Let F 2119864

rarr 119868 be a map Then the followingconditions are equivalent

(1) (119864F) is a fuzzifying greedoid(2) (119864F

[119886]) is a greedoid for each 119886 isin (0 1]

(3) (119864F(119886)) is a greedoid for each 119886 isin [0 1)

Definition 32 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

A mapping 119891 1198641

rarr 1198642is called an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2) if

F1(119891larr

119868(119860)) ge F

2(119860) for all 119860 isin 1198681198642 where 119891larr

119868(119860) = 119860 ∘ 119891

Definition 33 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

A mapping 119891 1198641rarr 1198642is called a fuzzifying feasibility pre-

serving mapping from (1198641F1) to (119864

2F2) ifF

1(119891minus1

(119860)) ge

F2(119860) for all 119860 isin 21198642

Theorem 34 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

and let 119891 1198641rarr 1198642be an (119868 119868)-fuzzy feasibility preserving

mapping from (1198641F1) to (119864

2F2) Then the following condi-

tions are equivalent

(1) 119891 is an (119868 119868)-fuzzy feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is an 119868-feasibility preserving mapping from(1198641 (F1)(119886)) to (119864

2 (F2)(119886)) for each 119886 isin [0 1)

Proof (1) rArr (2) Let119891 be an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1F1) to (119864

2F2) and 119886 isin (0 1] forall119860 isin

(F2)[119886] we have F

1(119891larr

119868(119860)) ge F

2(119860) ge 119886 and 119891larr

119868(119860) isin

(F1)[119886] In other words 119891 is a fuzzy feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin

(0 1](1) rArr (2) Next we need to prove that F

1(119891larr

119868(119860)) ge

F2(119860) for any 119860 isin 119868

119864 Let F2(119860) ge 119886 Then 119860 isin

(F2)[119886] Since 119891 is an 119868-feasibility preserving mapping from

(1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1] 119891larr

119868(119860) isin

(F1)[119886] that is F

1(119891larr

119868(119860)) ge 119886 By the arbitrariness of 119886

we haveF1(119891larr

119868(119860)) ge F

2(119860) Therefore 119891 is an (119868 119868)-fuzzy

feasibility preserving mapping from (1198641F1) to (119864

2F2)

Analogously we can obtain (1) hArr (3)

Corollary 35 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

and let119891 1198641rarr 1198642be a fuzzifying feasibility preservingmap-

ping from (1198641F1) to (119864

2F2) Then the following conditions

are equivalent

(1) 119891 is a fuzzifying feasibility preserving mapping from(1198641F1) to (119864

2F2)

(2) 119891 is a feasibility preserving mapping from (1198641 (F1)[119886])

to (1198642 (F2)[119886]) for each 119886 isin (0 1]

(3) 119891 is a feasibility preserving mapping from (1198641 (F1)(119886))

to (1198642 (F2)(119886)) for each 119886 isin [0 1)

Theorem 36 Let (119864F) be a fuzzifying greedoid Then forall119886 isin(0 1] (119864F[119886]) is a greedoid Since 119864 is a finite set there is at

6 Journal of Function Spaces

most a finite number of greedoids that can be defined on119864Thusthere is a finite sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

such that

(1) if 119887 119888 isin (119886119894minus1 119886119894] (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 le 119886

119894lt 119888 le 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]

The sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof For 119886 119887 isin (0 1] we can easily check that if 119886 le 119887 thenF[119886]supe F[119887] We define an equivalence relation sim on (0 1]

by 119886 sim 119887 hArr F[119886]= F[119887] Since 119864 is a finite set the number

of greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is

a left open and right closed interval It is trivial that each119868119894(119894 = 1 2 119899) is an interval Suppose that 119886

119894= sup 119868

119894

and (119864F119894) is the corresponding cut greedoid with respect

to 119868119894 Then F

119894supe F[119886119894]

holds For all 119860 isin F119894 119860 isin F

[119903]

and F(119860) ge 119903 for all 119903 isin 119868119894 It follows that F(119860) ge 119886

119894and

119860 isin F[119886119894]

Thus F119894sube F[119886119894]

Hence F119894= F[119886119894]

and 119886119894isin 119868119894

Obviously the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is

the fundamental sequence for (119864F)

5 FYG as a Subcategory of IG

In this section we will study the relation between fuzzifyinggreedoids and fuzzy greedoids from the viewpoint of categorytheory

Theorem 37 Let (119864F) be a fuzzifying greedoid and 120590(F) =119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then

(1) 120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) (119864 120590(F)) is a closed and perfect 119868-greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860[119886]isin 120590(F)[119886] where

119860 isin 120590(F) Then 119860[119886]isin F[119886]

by the definition of 120590(F) Thismeans that120590(F)[119886] sube F

[119886](forall119886 isin (0 1]) Conversely assume

that 119860 isin F[119886] It is obvious that (119886 and 119860)

[119887]= 119860 isin F

[119886]sube

F[119887]

for every 119886 ge 119887 and (119886 and 119860)[119887]= 0 isin F

[119887]for every

119886 lt 119887 Hence 119886 and 119860 isin 120590(F) So 119860 = (119886 and 119860)[119886]isin 120590(F)[119886]

This implies that 120590(F)[119886] supe F[119886](forall119886 isin (0 1]) Therefore

120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) It is easy to see that 120590(F) satisfies (IG1) Nowwe provethat 120590(F) satisfies (IG2) Suppose that 119860 119861 isin 120590(F) and|119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|[119887] We

know that |119861[119887]| gt |119860

[119887]| Since119860

[119887] 119861[119887]isin F[119886]

and (119864F[119887])

is a greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup 119890 isin

F[119887] In this case ((119887 and 119860

[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F

[119887] It is

obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F

[119887]sube F[119886]

for every 119886 le 119887 and ((119887 and 119860[119887]) cup 119890119887)[119886]= 0 isin F

[119886]for every

119886 gt 119887 This implies that (119887 and 119860[119887]) cup 119890119887isin 120590(F)

By (1) and the definition of 120590(F) 120590(F) = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

120590(F)[119886] By Corollary 22 (119864 120590(F)) is a perfect 119868-greedoidByTheorem 36 (119864 120590(F)) is a closed 119868-greedoid

Theorem 38 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an 119868-feasibility preserving

mapping from (1198641 120590(F1)) to (119864

2 120590(F2))

Proof Since 119891 is a fuzzifying feasibility preserving mappingfrom (119864

1F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

by Corollary 35 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120590(F1)[119886]) to (119864

2 120590(F2)[119886]) for each

119886 isin (0 1] Since (1198641 120590(F1)) is perfect by Theorem 24 119891

is an 119868-feasibility preserving mapping from (1198641 120590(F1)) to

(1198642 120590(F2))

Theorem 39 Let (119864F) be a closed and perfect 119868-greedoidDefine a map 120575(F) 2119864 rarr 119868 by

120575 (F) (119860) = ⋁119886 isin (0 1] 119860 isinF [119886] (11)

Then

(1) 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) (119864 120575(F)) is a fuzzifying greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860 isin F[119886] Then120575(F)(119860) ge 119886 by the definition of 120575(F) We have119860 isin 120575(F)

[119886]

This means that 120575(F)[119886]supe F[119886] Conversely assume that

119860 isin 120575(F)[119886] that is 120575(F)(119860) ge 119886 Since (119864F) is a closed 119868-

greedoid there exists 119887 isin (0 1] such that 119887 ge 119886 and119860 isin F[119887]By 119887 ge 119886 F[119886] supe F[119887] This means that 120575(F)

[119886]sube F[119886]

Therefore 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) By (1) and Corollary 31 (119864 120575(F)) is a fuzzifyinggreedoid

Theorem 40 Let (119864119894F119894) (119894 = 1 2) be closed and perfect 119868-

greedoids If 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) then119891 is a fuzzifying feasibility

preserving mapping from (1198641 120575(F1)) to (119864

2 120575(F2))

Proof Since 119891 is an 119868-feasibility preserving mapping from(1198641F1) to (119864

2F2) 119891 is a feasibility preserving mapping

from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1] by

Theorem 24 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120575(F1)[119886]) to (119864

2 120575(F2)[119886]) for each 119886 isin

(0 1] By Corollary 35 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120575(F1)) to (119864

2 120575(F2))

Theorem 41 (1) If (119864F) is a fuzzifying greedoid then 120575 ∘120590(F) = F

(2) If (119864F) is a closed and perfect 119868-greedoid then 120590 ∘120575(F) = F

Proof (1) ByTheorem 37 (119864 120590(F)) is a closed and perfect 119868-greedoidThen 120575∘120590(F)

[119886]= 120590(F)[119886] = F

[119886]for all 119886 isin (0 1]

byTheorems 37 and 39 Hence 120575 ∘ 120590(F) = F(2) By Theorems 37 and 39 (119864 120590 ∘ 120575(F)) is a closed and

perfect 119868-greedoid Then 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1]119860[119886]

isin 120575(F)[119886] Since (119864F) is a closed and perfect 119868-

greedoid 120575(F)[119886]= F[119886] for all 119886 isin (0 1] by Theorem 39

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

most a finite number of greedoids that can be defined on119864Thusthere is a finite sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

such that

(1) if 119887 119888 isin (119886119894minus1 119886119894] (1 le 119894 le 119903) thenF[119887] = F[119888]

(2) if 119886119894minus1lt 119887 le 119886

119894lt 119888 le 119886

119894+1(1 le 119894 le 119903 minus 1) then

F[119887] ⫌ F[119888]

The sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is called the

fundamental sequence for (119864F)

Proof For 119886 119887 isin (0 1] we can easily check that if 119886 le 119887 thenF[119886]supe F[119887] We define an equivalence relation sim on (0 1]

by 119886 sim 119887 hArr F[119886]= F[119887] Since 119864 is a finite set the number

of greedoids on 119864 is finite Thus there exist at most finitelymany equivalence classes which are respectively denoted by1198681 1198682 119868

119899 Next we prove that each 119868

119894(119894 = 1 2 119899) is

a left open and right closed interval It is trivial that each119868119894(119894 = 1 2 119899) is an interval Suppose that 119886

119894= sup 119868

119894

and (119864F119894) is the corresponding cut greedoid with respect

to 119868119894 Then F

119894supe F[119886119894]

holds For all 119860 isin F119894 119860 isin F

[119903]

and F(119860) ge 119903 for all 119903 isin 119868119894 It follows that F(119860) ge 119886

119894and

119860 isin F[119886119894]

Thus F119894sube F[119886119894]

Hence F119894= F[119886119894]

and 119886119894isin 119868119894

Obviously the sequence 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1 is

the fundamental sequence for (119864F)

5 FYG as a Subcategory of IG

In this section we will study the relation between fuzzifyinggreedoids and fuzzy greedoids from the viewpoint of categorytheory

Theorem 37 Let (119864F) be a fuzzifying greedoid and 120590(F) =119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Then

(1) 120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) (119864 120590(F)) is a closed and perfect 119868-greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860[119886]isin 120590(F)[119886] where

119860 isin 120590(F) Then 119860[119886]isin F[119886]

by the definition of 120590(F) Thismeans that120590(F)[119886] sube F

[119886](forall119886 isin (0 1]) Conversely assume

that 119860 isin F[119886] It is obvious that (119886 and 119860)

[119887]= 119860 isin F

[119886]sube

F[119887]

for every 119886 ge 119887 and (119886 and 119860)[119887]= 0 isin F

[119887]for every

119886 lt 119887 Hence 119886 and 119860 isin 120590(F) So 119860 = (119886 and 119860)[119886]isin 120590(F)[119886]

This implies that 120590(F)[119886] supe F[119886](forall119886 isin (0 1]) Therefore

120590(F)[119886] = F[119886](forall119886 isin (0 1])

(2) It is easy to see that 120590(F) satisfies (IG1) Nowwe provethat 120590(F) satisfies (IG2) Suppose that 119860 119861 isin 120590(F) and|119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|[119887] We

know that |119861[119887]| gt |119860

[119887]| Since119860

[119887] 119861[119887]isin F[119886]

and (119864F[119887])

is a greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup 119890 isin

F[119887] In this case ((119887 and 119860

[119887]) cup 119890119887)[119887]= 119860[119887]cup 119890 isin F

[119887] It is

obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]= 119860[119887]cup 119890 isin F

[119887]sube F[119886]

for every 119886 le 119887 and ((119887 and 119860[119887]) cup 119890119887)[119886]= 0 isin F

[119886]for every

119886 gt 119887 This implies that (119887 and 119860[119887]) cup 119890119887isin 120590(F)

By (1) and the definition of 120590(F) 120590(F) = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

120590(F)[119886] By Corollary 22 (119864 120590(F)) is a perfect 119868-greedoidByTheorem 36 (119864 120590(F)) is a closed 119868-greedoid

Theorem 38 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an 119868-feasibility preserving

mapping from (1198641 120590(F1)) to (119864

2 120590(F2))

Proof Since 119891 is a fuzzifying feasibility preserving mappingfrom (119864

1F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641 (F1)[119886]) to (119864

2 (F2)[119886]) for each 119886 isin (0 1]

by Corollary 35 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120590(F1)[119886]) to (119864

2 120590(F2)[119886]) for each

119886 isin (0 1] Since (1198641 120590(F1)) is perfect by Theorem 24 119891

is an 119868-feasibility preserving mapping from (1198641 120590(F1)) to

(1198642 120590(F2))

Theorem 39 Let (119864F) be a closed and perfect 119868-greedoidDefine a map 120575(F) 2119864 rarr 119868 by

120575 (F) (119860) = ⋁119886 isin (0 1] 119860 isinF [119886] (11)

Then

(1) 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) (119864 120575(F)) is a fuzzifying greedoid

Proof (1) forall119886 isin (0 1] suppose that 119860 isin F[119886] Then120575(F)(119860) ge 119886 by the definition of 120575(F) We have119860 isin 120575(F)

[119886]

This means that 120575(F)[119886]supe F[119886] Conversely assume that

119860 isin 120575(F)[119886] that is 120575(F)(119860) ge 119886 Since (119864F) is a closed 119868-

greedoid there exists 119887 isin (0 1] such that 119887 ge 119886 and119860 isin F[119887]By 119887 ge 119886 F[119886] supe F[119887] This means that 120575(F)

[119886]sube F[119886]

Therefore 120575(F)[119886]= F[119886] for all 119886 isin (0 1]

(2) By (1) and Corollary 31 (119864 120575(F)) is a fuzzifyinggreedoid

Theorem 40 Let (119864119894F119894) (119894 = 1 2) be closed and perfect 119868-

greedoids If 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) then119891 is a fuzzifying feasibility

preserving mapping from (1198641 120575(F1)) to (119864

2 120575(F2))

Proof Since 119891 is an 119868-feasibility preserving mapping from(1198641F1) to (119864

2F2) 119891 is a feasibility preserving mapping

from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1] by

Theorem 24 By Theorem 37 119891 is a feasibility preservingmapping from (119864

1 120575(F1)[119886]) to (119864

2 120575(F2)[119886]) for each 119886 isin

(0 1] By Corollary 35 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120575(F1)) to (119864

2 120575(F2))

Theorem 41 (1) If (119864F) is a fuzzifying greedoid then 120575 ∘120590(F) = F

(2) If (119864F) is a closed and perfect 119868-greedoid then 120590 ∘120575(F) = F

Proof (1) ByTheorem 37 (119864 120590(F)) is a closed and perfect 119868-greedoidThen 120575∘120590(F)

[119886]= 120590(F)[119886] = F

[119886]for all 119886 isin (0 1]

byTheorems 37 and 39 Hence 120575 ∘ 120590(F) = F(2) By Theorems 37 and 39 (119864 120590 ∘ 120575(F)) is a closed and

perfect 119868-greedoid Then 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1]119860[119886]

isin 120575(F)[119886] Since (119864F) is a closed and perfect 119868-

greedoid 120575(F)[119886]= F[119886] for all 119886 isin (0 1] by Theorem 39

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 7

Therefore 120590 ∘ 120575(F) = 119860 isin 119868119864 forall119886 isin (0 1] 119860[119886]isin F[119886] =

F since (119864F) is a closed and perfect 119868-greedoid

ByTheorems 37ndash41 we have the following

Theorem 42 FYG is isomorphic to CPIG

Theorem 43 Let (119864F) be an 119868-greedoid with the funda-mental sequence 0 = 119886

0lt 1198861lt sdot sdot sdot lt 119886

119903minus1lt 119886119903= 1

forall119886 isin (0 1] define F119886= F[(119886

119894minus1+ 119886119894)2] where 119886 isin (119886

119894minus1 119886119894]

Let F = 119860 isin 119868119864

forall119886 isin (0 1] 119860[119886]isin F119886 Then

(1) (119864 F) is an 119868-greedoid

(2) F[119886] = F119886for all 119886 isin (0 1]

(3) (119864 F) is a closed and perfect 119868-greedoid

Proof (1) It is easy to see that F satisfies (IG1) Nowwe provethat F satisfies (IG2) Suppose that 119860 119861 isin F and |119861|(119899) ge119887 gt |119860|(119899) for some 119899 isin N Then |119861|

[119887]gt |119860|

[119887] We know

that |119861[119887]| gt |119860

[119887]| Since 119860

[119887] 119861[119887]isin F119887and (119864 F

119887) is a

greedoid there exists 119890 isin 119861[119887]minus119860[119887]

such that119860[119887]cup119890 isin F

119887

In this case 119890 isin 119865(119860 119861) It is obvious that ((119887and119860[119887])cup119890119887)[119886]=

119860[119887]cup119890 isin F

119887sube F119886for every 119886 le 119887 and ((119887and119860

[119887])cup119890119887)[119886]=

0 isin F119886for every 119886 gt 119887 This implies that (119887 and 119860

[119887]) cup 119890119887isin F

(2) Suppose that 119860 isin F[119886] Then there exists 119860 isin F

such that 119860 = 119860[119886]isin F[119886] and 119860

[119887]isin F119887for each 119887 isin

(0 1] by the definition of F This implies that F[119886] sube F119886

Conversely assume that 119860 isin F119886 Then there exists 119860 isin F

such that 119860[119886]= 119860 and 119886 and119860

[119886]isin F We have (119886 and 119860

[119886])[119887]=

119860[119886]isin F119886sube F119887for every 119886 ge 119887 and (119886 and 119860

[119886])[119887]= 0 isin F

119887

for every 119887 gt 119886 This implies that 119886 and 119860[119886]isin F Hence 119860 =

119860[119886]= (119886 and 119860

[119886])[119886]isin F[119886] We have F

119886sube F[119886] Therefore

F[119886] = F119886for all 119886 isin (0 1]

(3) By (1) and (2) (119864 F) is a closed 119868-greedoid By (2)and the definition of F F = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin

F119886 = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] Hence (119864 F) is

a perfect 119868-greedoid

Theorem 44 If (119864F) is a closed and perfect 119868-greedoid thenF = F

Proof Since (119864F) is a closed 119868-greedoid by Theorem 43F[119886] = F[119886] for all 119886 isin (0 1] By Theorem 43 (119864 F) is aperfect 119868-greedoid Since (119864F) is a perfect 119868-greedoid thenF = 119860 isin 119868

119864

forall119886 isin (0 1] 119860[119886]isin F[119886] = 119860 isin 119868119864 forall119886 isin

(0 1] 119860[119886]isin F[119886] = F

Theorem 45 Let (119864119894F119894) (119894 = 1 2) be 119868-greedoids If 119891

1198641rarr 1198642is an 119868-feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 F1) to (119864

2 F2)

Proof Since 119891 1198641rarr 1198642is an 119868-feasibility preserving map-

ping from (1198641F1) to (119864

2F2) 119891 is a feasibility preserving

mapping from (1198641F1[119886]) to (119864

2F2[119886]) for each 119886 isin (0 1]

by Theorem 24 Let 0 = 1198860lt 1198861lt sdot sdot sdot lt 119886

119899minus1lt 119886119899= 1

and 0 = 1198870lt 1198871lt sdot sdot sdot lt 119887

119898minus1lt 119887119898= 1 be the fundamental

sequences for (1198641F1) and (119864

2F2) respectively forall119886 isin (0 1]

there exist 119894 isin 1 2 119899 and 119895 isin 1 2 119898 such that 119886 isin(119886119894minus1 119886119894] cap (119887119895minus1 119887119895] Let 119888 = (12)min119886minus119886

119894minus1 119886 minus 119887119895minus1 Then

119886 minus 119888 isin (119886119894minus1 119886119894] cap (119887119895minus1 119887119895] It is easy to verify that F

1[119886] =

F1[119886 minus 119888] and F

2[119886] = F

2[119886 minus 119888] Thus 119891 is an 119868-feasibility

preserving mapping from (1198641 F1[119886]) to (119864

2 F2[119886]) for all

119886 isin (0 1] By Theorem 43 (1198641 F1) is a perfect 119868-greedoid

ByTheorem 24119891 is an 119868-feasibility preservingmapping from(1198641 F1) to (119864

2 F2)

Therefore there is a functor from IG to CPIG sendingevery 119868-greedoid (119864F) to (119864 F) in symbols 119904

By Theorems 43ndash45 we have the following

Theorem 46 Let 119894 CPIG rarr IG be the inclusion functorThen

(1) 119894 ∘ 119904(F) supe F for any 119868-greedoid (119864F)(2) 119904 ∘ 119894(F) = F for any closed and perfect 119868-greedoid(119864F)

ByTheorem 46 we have the following

Theorem 47 CPIG is a coreflective full subcategory of IG

ByTheorems 47 and 42 we have the following

Corollary 48 FYG can be regarded as a coreflective fullsubcategory of IG

6 G as a Subcategory of FYG and IG asa Subcategory of IIFG

In the following we will study the relation between 119868-greedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and fuzzifying greedoids from the viewpoint ofcategory theory

Let (119864F) be an 119868-greedoid Then (119864 120594F) is an (119868 119868)-fuzzy greedoid Therefore we define the inclusion mapping119894 IG rarr IIFG by 119894(119864F) = (119864 120594F)

By Theorem 30 Corollary 31 Theorem 34 andCorollary 35 we will define two functors from IIFG to IG inthe following theorem

Theorem 49 (1) 119865 IIFG rarr IG (119864F) 997891rarr (119864F[1]) is a

functor(2) 119866 IIFG rarr IG (119864F) 997891rarr (119864F

(0)) is a functor

Theorem 50 (1) For any 119868-greedoid (119864F) 119865 ∘ 119894(F) = 119866 ∘119894(F) = F

(2) For any (119868 119868)-fuzzy greedoid (119864F) 119894 ∘ 119865(F) le F and119894 ∘ 119866(F) ge F

Proof The proof is trivial and straightforward

Since both IG and IIFG are concrete categories we havethe following

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces

Theorem 51 Both (119865 119894) and (119894 119866) are Galois correspondencesbetween IG and IIFG and 119894 is the right inverse of both 119865 and119866 Hence IG can be embedded in IIFG as a simultaneouslyreflective and coreflective full subcategory

Corollary 52 G is a simultaneously reflective and coreflectivefull subcategory of FYG

7 G as a Subcategory of IG and FYG asa Subcategory of IIFG

In this section we will study the relation between fuzzifyinggreedoids and (119868 119868)-fuzzy greedoids and the relation betweengreedoids and 119868-greedoids from the viewpoint of categorytheory

Theorem 53 Let (119864F) be a fuzzifying greedoid Define120596119868(F) 119868119864 rarr 119868 by

120596119868(F) (119860) = ⋀

119886isin(01]

F (119860[119886]) (12)

Then (119864 120596119868(F)) is an (119868 119868)-fuzzy greedoid

Proof It is easy to see that 120596119868(F) satisfies (IIFG1) Now

we prove that 120596119868(F) satisfies (IIFG2) Suppose that 119860 119861 isin

[0 1]119864 and |119861|(119899) ge 119887 gt |119860|(119899) for some 119899 isin N Then

|119861|[119887]gt |119860|

[119887] We know that |119861

[119887]| gt |119860

[119887]| This implies

⋁119890isin119861[119887]minus119860[119887]

F(119860[119887]cup 119890) ge F(119860

[119887]) andF(119861

[119887]) since (119864F) is

a fuzzifying greedoid It is obvious that ((119887 and 119860[119887]) cup 119890119887)[119886]=

119860[119887]cup 119890 for every 119886 le 119887 and ((119887 and119860

[119887]) cup 119890119887)[119886]= 0 for every

119886 gt 119887 This implies

⋁

119890isin119865(119860119861)

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

120596119868(F) ((119887 and 119860

[119887]) cup 119890119887)

ge ⋁

119890isin119861[119887]minus119860[119887]

⋀

119886isin(01]

F (((119887 and 119860[119887]) cup 119890119887)[119886])

= ⋁

119890isin119861[119887]minus119860[119887]

F (119860[119887]cup 119890) andF (0)

ge F (119860[119887]) andF (119861

[119887])

ge 120596119868(F) (119860

[119887]) and 120596119868(F) (119861

[119887])

(13)

where 119865(119860 119861) = 119890 isin 119864 119861(119890) ge 119887 gt 119860(119890) Therefore (119864120596119868(F)) is an (119868 119868)-fuzzy greedoid

Corollary 54 Let (119864F) be a greedoid Define 120596(F) by120596(F) = 119860 isin 119868119864 119860

[119886]isin F forall119886 isin (0 1] Then (119864 120596(F)) is

an 119868-greedoid

Theorem 55 Let (119864119894F119894) (119894 = 1 2) be fuzzifying greedoids

If 119891 1198641rarr 1198642is a fuzzifying feasibility preserving mapping

from (1198641F1) to (119864

2F2) then 119891 is an (119868 119868)-fuzzy feasibility

preserving mapping from (1198641 120596119868(F1)) to (119864

2 120596119868(F2))

Proof forall119860 isin 119868119864 119860[119886]isin 2119864 for all 119886 isin (0 1] Since 119891 119864

1rarr

1198642is a fuzzifying feasibility preserving mapping from (119864

1

F1) to (119864

2F2)F1(119891minus1

(119860[119886])) ge F

2(119860[119886]) for all 119886 isin (0 1]

By the definition of 120596119868(F1) and 120596

119868(F2) we have

120596119868(F1) (119891larr

119868(119860)) = ⋀

119886isin(01]

F1((119891larr

119868(119860))[119886])

= ⋀

119886isin(01]

F1((119891minus1

(119860[119886])))

ge ⋀

119886isin(01]

F2(119860[119886]) = 120596119868(F2) (119860)

(14)

This implies that 119891 is an (119868 119868)-fuzzy feasibility preservingmapping from (119864

1 120596119868(F1)) to (119864

2 120596119868(F2))

Corollary 56 Let (119864119894F119894) (119894 = 1 2) be greedoids If 119891

1198641rarr 119864

2is a feasibility preserving mapping from (119864

1F1)

to (1198642F2) then 119891 is an 119868-feasibility preserving mapping from

(1198641 120596(F

1)) to (119864

2 120596(F

2))

Theorem 57 Let (119864F) be an (119868 119868)-fuzzy greedoid Define120580119868(F) 2119864 rarr 119868 by

120580119868(F) (119860) = ⋁

119886isin(01]

F (119860 and 119886) (15)

Then (119864 120580119868(F)) is a fuzzifying greedoid

Proof It is easy to see that 120580119868(F) satisfies (FYG1) Now we

prove that 120580119868(F) satisfies (FYG2) Suppose that119860 119861 isin 2119864 and

|119860| lt |119861| Then 119886 and 119860 119886 and 119861 isin 119868119864 and |119886 and 119861|(|119861|) = 119886 gt 0 =|119886 and 119860|(|119861|) for each 119886 isin (0 1] Since (119864F) is an (119868 119868)-fuzzygreedoid

⋁

119890isin119865(119886and119860119886and119861)

F ((119886 and (119860 and 119886)[119886]) or 119890119886)

ge F (119860 and 119886) andF (119861 and 119886)

(16)

that is

⋁

119890isin119861minus119860

F (119886 and (119860 cup 119890)) ge F (119860 and 119886) andF (119861 and 119886) (17)

Hence

⋁

119890isin119861minus119860

120580119868(F) (119886 and (119860 cup 119890))

= ⋁

119890isin119861minus119860

⋁

119886isin(01]

F (119886 and (119860 cup 119890))

ge ⋁

119886isin(01]

F (119860 and 119886) and ⋁

119886isin(01]

F (119861 and 119886)

= 120580119868(F) (119860) and 120580

119868(F) (119861)

(18)

This implies that (FYG2) holds Therefore (119864 120580119868(F)) is a

fuzzifying greedoid

Corollary 58 Let (119864F) be an 119868-greedoid Define 120580(F) by120580(F) = 119860 isin 2

119864

119860 and 119886 isin F for some 119886 isin (0 1] Then(119864F) is a greedoid

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 9

Theorem 59 Let (119864119894F119894) (119894 = 1 2) be (119868 119868)-fuzzy greedoids

If119891 1198641rarr 1198642is an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) then 119891 is a fuzzifying feasibility

preserving mapping from (1198641 120580119868(F1)) to (119864

2 120580119868(F2))

Proof For each 119860 isin 2119864 by the definition of 120580

119868(F) and 119891

1198641rarr 1198642being an (119868 119868)-fuzzy feasibility preservingmapping

from (1198641F1) to (119864

2F2) we have

120580119868(F1) (119891minus1

(119860)) = ⋁

119886isin(01]

F1(119891minus1

(119860) and 119886)

= ⋁

119886isin(01]

F1(119891larr

119868(119860 and 119886))

ge ⋁

119886isin(01]

F2(119860 and 119886) = 120580

119868(F2) (119860)

(19)

This implies that 119891 is a fuzzifying feasibility preservingmapping from (119864

1 120580119868(F1)) to (119864

2 120580119868(F2))

Corollary 60 Let (119864119894F119894) (119894 = 1 2) be fuzzy greedoids If

119891 1198641rarr 119864

2is an 119868-feasibility preserving mapping from

(1198641F1) to (119864

2F2) then119891 is a feasibility preserving mapping

from (1198641 120580(F1)) to (119864

2 120580(F2))

Theorem 61 (1) If (119864F) is an (119868 119868)-fuzzy greedoid then 120596119868∘

120580119868(F) ge F(2) If (119864F) is a fuzzifying greedoid then 120580

119868∘ 120596119868(F) = F

Proof (1) For each 119860 isin 119868119864 by the definitions of 120580119868and 120596

119868 we

have

120596119868∘ 120580119868(F) (119860) = ⋀

119886isin(01]

120580119868(F) (119860

[119886])

= ⋀

119886isin(01]

⋁

119887isin(01]

F (119887 and 119860[119886])

ge ⋀

119886isin(01]

F (119886 and 119860[119886]) ge F (119860)

(20)

This implies that 120596119868∘ 120580119868(F) ge F

(2) For each 119860 isin 2119864 by the definitions of 120580119868and 120596

119868 we

have

120580119868∘ 120596119868(F) (119860) = ⋁

119886isin(01]

120596119868(F) (119886 and 119860)

= ⋁

119886isin(01]

⋀

119887isin(01]

F ((119886 and 119860)[119887])

= ⋁

119886isin(01]

F (119860) = F (119860)

(21)

This implies that 120580119868∘ 120596119868(F) = F

Corollary 62 (1) If (119864F) is an 119868-greedoid then120596∘120580(F) supe F(2) If (119864F) is a greedoid then 120580 ∘ 120596(F) = F

Based on the theorems and corollaries in this section wehave the following

Theorem 63 (1) FYG is a coreflective full subcategory ofIIFG

(2) G is a coreflective full subcategory of IG

8 Conclusion

In this paper we introduce the concepts of 119868-greedoidsfuzzifying greedoids and (119868 119868)-fuzzy greedoids as differentapproaches of the fuzzification of greedoids and study therelations among greedoids fuzzy greedoids fuzzifying gree-doids and (119868 119868)-fuzzy greedoids from a categorical point ofview The main results in this paper are in the followingdiagram where 119903 119888 mean respectively reflective and core-flective

IIFG

IG CPIGc iso FYG

c

Gc

r c

r c

(22)

Such facts will be useful to help further investigations and it ispossible that the fuzzification of greedoids would be appliedto some combinatorial optimization problems in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project is supported by the National Natural ScienceFoundation of China (11371002) and Specialized ResearchFund for the Doctoral Program of Higher Education(20131101110048)

References

[1] B Korte and L Lovasz ldquoMathemalical structures underlyinggreedy algorithmsrdquo in Fundamentals of Computation Theory(Szeged 1981) vol 117 of Lecture Notes in Computer Science pp205ndash209 Springer 1981

[2] B Korte and L Lovasz ldquoGreedoidsmdasha structural framework forthe greedy algorithmrdquo in Progress in Combinatorial Optimiza-tion Proceedings of the Silver Jubillee Conference on Combina-torics (Waterloo June 1982) R Pulleyblank Ed pp 221ndash243Academic Press London UK 1984

[3] J Goetschel and W Voxman ldquoFuzzy matroidsrdquo Fuzzy Sets andSystems vol 27 no 3 pp 291ndash302 1988

[4] J Goetschel and W Voxman ldquoBases of fuzzy matroidsrdquo FuzzySets and Systems vol 31 no 2 pp 253ndash261 1989

[5] R Goetschel Jr and W Voxman ldquoFuzzy matroids and a greedyalgorithmrdquo Fuzzy Sets and Systems vol 37 no 2 pp 201ndash2131990

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Function Spaces

[6] R Goetschel Jr and W Voxman ldquoFuzzy rank functionsrdquo FuzzySets and Systems vol 42 no 2 pp 245ndash258 1991

[7] Y C Hsueh ldquoOn fuzzification of matroidsrdquo Fuzzy Sets andSystems vol 53 no 3 pp 319ndash327 1993

[8] S-G Li X Xin and Y-L Li ldquoClosure axioms for a class of fuzzymatroids and co-towers of matroidsrdquo Fuzzy Sets and Systemsvol 158 no 11 pp 1246ndash1257 2007

[9] X-N Li S-Y Liu and S-G Li ldquoConnectedness of refinedGoetschel-Voxman fuzzymatroidsrdquo Fuzzy Sets and Systems vol161 no 20 pp 2709ndash2723 2010

[10] L A Novak ldquoA comment on lsquoBases of fuzzy matroidsrsquordquo FuzzySets and Systems vol 87 no 2 pp 251ndash252 1997

[11] L A Novak ldquoOn fuzzy independence set systemsrdquo Fuzzy Setsand Systems vol 91 no 3 pp 365ndash374 1997

[12] L ANovak ldquoOnGoetschel andVoxman fuzzymatroidsrdquo FuzzySets and Systems vol 117 no 3 pp 407ndash412 2001

[13] F-G Shi ldquoA new approach to the fuzzification of matroidsrdquoFuzzy Sets and Systems vol 160 no 5 pp 696ndash705 2009

[14] H Lian and X Xin ldquoThe nullities for M-fuzzifying matroidsrdquoApplied Mathematics Letters vol 25 no 3 pp 279ndash286 2012

[15] F-G Shi and L Wang ldquoCharacterizations and applications ofM-fuzzifying matroidsrdquo Journal of Intelligent amp Fuzzy Systemsvol 25 no 4 pp 919ndash930 2013

[16] L Wang and Y-P Wei ldquoM-fuzzifying P-closure operatorsrdquoin Fuzzy Information and Engineering Volume 2 vol 62 ofAdvances in Intelligent and Soft Computing pp 547ndash554Springer Berlin Germany 2009

[17] W Yao and F-G Shi ldquoBases axioms and circuits axioms forfuzzifyingmatroidsrdquo Fuzzy Sets and Systems vol 161 no 24 pp3155ndash3165 2010

[18] L-X Lu and W-W Zheng ldquoCategorical relations amongmatroids fuzzymatroids and fuzzifyingmatroidsrdquo Iranian Jour-nal of Fuzzy Systems vol 7 no 1 pp 81ndash89 2010

[19] F-G Shi ldquo(L M)-fuzzy matroidsrdquo Fuzzy Sets and Systems vol160 no 16 pp 2387ndash2400 2009

[20] C-E Huang and F-G Shi ldquoBases of L-matroidsrdquo HacettepeJournal of Mathematics and Statistics vol 39 pp 233ndash240 2010

[21] H Lian and X Xin ldquo[0 1]-fuzzy 120573-rank mappingsrdquo HacettepeJournal of Mathematics and Statistics vol 41 no 1 pp 59ndash652012

[22] X Xin and F-G Shi ldquoCategories of bi-fuzzy pre-matroidsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1548ndash1558 2010

[23] X Xin and F-G Shi ldquoRank functions for closed and perfect[0 1]-matroidsrdquoHacettepe Journal ofMathematics and Statisticsvol 39 no 1 pp 31ndash39 2010

[24] J G OxleyMatroidTheory Oxford University Press New YorkNY USA 1992

[25] D J A Welsh Matroid Applicationa Cambridge UniversityPress New York NY USA 1992

[26] M L J Van de VelTheory of Convex Structures North-HollandNew York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of