Research Article -Polar Fuzzy Sets: An Extension of Bipolar Fuzzy …downloads.hindawi.com/journals/tswj/2014/416530.pdf · 2019-07-31 · Research Article-Polar Fuzzy Sets: An Extension

Research Article119898-Polar Fuzzy Sets An Extension of Bipolar Fuzzy Sets

Juanjuan Chen12 Shenggang Li1 Shengquan Ma1 and Xueping Wang3

1 College of Mathematics and Information Science Shaanxi Normal University Xirsquoan 710062 China2 School of Sciences Xirsquoan University of Technology Xirsquoan 710056 China3 College of Mathematics and Software Science Sichuan Normal University Chengdu 610066 China

Correspondence should be addressed to Shenggang Li shengganglinew126com

Received 11 April 2014 Accepted 23 May 2014 Published 12 June 2014

Academic Editor Jianming Zhan

Copyright copy 2014 Juanjuan Chen et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recently bipolar fuzzy sets have been studied and applied a bit enthusiastically and a bit increasingly In this paper we provethat bipolar fuzzy sets and [0 1]2-sets (which have been deeply studied) are actually cryptomorphic mathematical notions Sinceresearches or modelings on real world problems often involve multi-agent multi-attribute multi-object multi-index multi-polarinformation uncertainty orand limit process we put forward (or highlight) the notion of 119898-polar fuzzy set (actually [0 1]119898-setwhich can be seen as a generalization of bipolar fuzzy set where119898 is an arbitrary ordinal number) and illustrate howmany conceptshave been defined based on bipolar fuzzy sets and many results which are related to these concepts can be generalized to the caseof119898-polar fuzzy sets We also give examples to show how to apply119898-polar fuzzy sets in real world problems

1 Introduction and Preliminaries

Set theory and logic systems are strongly coupled in thedevelopment of modern logic Classical logic corresponds tothe crisp set theory and fuzzy logic is associated with fuzzyset theory which was proposed by Zadeh in his pioneer work[1]

Definition 1 An 119871-subset (or an 119871-set) on the set 119883 is asynonym of a mapping 119860 119883 rarr 119871 where 119871 is a lattice (cf[2]) When 119871 = [0 1] (the ordinary closed unit interval withthe ordinary order relation) an 119871-set on 119883 will be called afuzzy set on119883 (cf [1])

The theory of fuzzy sets has become a vigorous area ofresearch in different disciplines including medical and lifesciences management sciences social sciences engineeringstatistics graph theory artificial intelligence pattern recog-nition robotics computer networks decision making andautomata theory

An extension of fuzzy set called bipolar fuzzy set wasgiven by Zhang [3] in 1994

Definition 2 (see Zhang [3]) A bipolar fuzzy set is a pair(120583+ 120583minus) where 120583+ 119883 rarr [0 1] and 120583minus 119883 rarr [minus1 0] are

anymappingsThe set of all bipolar fuzzy sets on119883 is denotedby BF(119883)

Bipolar fuzzy sets are an extension of fuzzy sets whosemembership degree range is [minus1 1] In a bipolar fuzzy set themembership degree 0 of an element means that the elementis irrelevant to the corresponding property the member-ship degree (0 1] of an element indicates that the elementsomewhat satisfies the property and the membership degree[minus1 0) of an element indicates that the element somewhatsatisfies the implicit counter-property The idea which liesbehind such description is connected with the existence ofldquobipolar informationrdquo (eg positive information and nega-tive information) about the given set Positive informationrepresents what is granted to be possible while negativeinformation represents what is considered to be impossibleActually a wide variety of human decision making is basedon double-sided or bipolar judgmental thinking on a positiveside and a negative side For instance cooperation andcompetition friendship and hostility common interests andconflict of interests effect and side effect likelihood andunlikelihood feedforward and feedback and so forth areoften the two sides in decision and coordination In thetraditional Chinese medicine (TCM for short) ldquoyinrdquo and

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 416530 8 pageshttpdxdoiorg1011552014416530

2 The Scientific World Journal

ldquoyangrdquo are the two sides Yin is the feminine or negative sideof a system and yang is the masculine or positive side of asystem The coexistence equilibrium and harmony of thetwo sides are considered a key for the mental and physicalhealth of a person as well as for the stability and prosperity ofa social systemThus bipolar fuzzy sets indeed have potentialimpacts on many fields including artificial intelligencecomputer science information science cognitive sciencedecision science management science economics neuralscience quantum computing medical science and socialscience (cf [4ndash45]) In recent years bipolar fuzzy sets seemto have been studied and applied a bit enthusiastically and abit increasingly (cf [4ndash45]) This is the chief motivation forus to introduce and study119898-polar fuzzy sets

The first object of this note is to answer the followingquestion on bipolar fuzzy sets

Question 1 Is bipolar fuzzy set a very intuitive 119871-setThe answer to Question 1 is positive We will prove in

this note that there is a natural one-to-one correspondencebetween BF(119883) and 2(119883) (for the set of all [0 1]2-sets on119883 see Theorem 5) which preserves all involved propertiesThis makes the notion of bipolar fuzzy set more intuitiveSince properties of 119871-sets have already been studied verydeeply and exhaustively this one-to-one correspondencemay be beneficial for both researchers interested in above-mentioned papers and related fields (because they can usethese properties directly and even cooperate with theoreticalfuzzy mathematicians for a possible higher-level research)and theoretical fuzzy mathematicians as well (because coop-eration with applied fuzzy mathematicians and practitionersprobably makes their research more useful)

We notice that ldquomultipolar informationrdquo (not just bipolarinformation which corresponds to two-valued logic) existsbecause data for a real world problem are sometimes from119899 agents (119899 ge 2) For example the exact degree oftelecommunication safety ofmankind is a point in [0 1]119899 (119899 asymp7times109) because different person has beenmonitored different

timesThere aremany other examples truth degrees of a logicformula which are based on 119899 logic implication operators(119899 ge 2) similarity degrees of two logic formulas which arebased on 119899 logic implication operators (119899 ge 2) orderingresults of a magazine ordering results of a university andinclusion degrees (resp accuracy measures roughmeasuresapproximation qualities fuzziness measures and decisionperformance evaluations) of a rough set Thus our secondobject of this note is to answer the following question onextensions of bipolar fuzzy sets

Question 2 How to generalize bipolar fuzzy sets to multipo-lar fuzzy sets and how to generalize results on bipolar fuzzysets to the case of multipolar fuzzy sets

The idea to answer Question 2 is from the answer toQuestion 1 intuitiveness of the point-wise order on [0 1]

119898

(see Remark 3) and the proven corresponding results onbipolar fuzzy sets We put forward the notion of 119898-polarfuzzy set (an extension of bipolar fuzzy set) and point outthatmany concepts which have been defined based on bipolarfuzzy sets and many results related to these concepts can be

y

x

u

vu

uv

y y

v

x x

Figure 1

y y

v

vu

u

x x

Figure 2

generalized to the case of 119898-polar fuzzy sets (see Remarks 7and 8 for details)

Apart from the backgrounds (eg ldquomultipolar informa-tionrdquo) of119898-polar fuzzy sets the following question on furtherapplications (particularly further applications in real worldproblems) of119898-polar fuzzy sets should also be considered

Question 3 How to find further possible applications of 119898-polar fuzzy sets in real world problems

Question 3 can be answered as in the case of bipolar fuzzysets since researches or modelings on real world problemsoften involve multiagent multiattribute multiobject multi-index multipolar information uncertainty orand limitsprocess We will give examples to demonstrate it (see Exam-ples 9ndash14)

Remark 3 In this note [0 1]119898 (119898-power of [0 1]) is con-sidered a poset with the point-wise order le where 119898 is anarbitrary ordinal number (wemake an appointment that119898 =

119899 | 119899 lt 119898 when 119898 gt 0) le (which is actually very intuitiveas illustrated below) is defined by 119909 le 119910 hArr 119901

119894(119909) le 119901

119894(119910) for

each 119894 isin 119898 (119909 119910 isin [0 1]119898) and 119901

119894 [0 1]

119898rarr [0 1] is the

119894th projection mapping (119894 isin 119898)

(1) When119898 = 2 [0 1]2 is the ordinary closed unit squarein Euclidean plane 1198772 The righter (resp the upper) apoint in this square is the larger it is Let 119909 = ⟨0 0⟩ =0 (the smallest element of [0 1]2) 119906 = ⟨025 075⟩V = ⟨075 025⟩ and 119910 = ⟨1 1⟩ (the largest element of[0 1]2) Then 119909 le 119911 le 119910 for all 119911 isin [0 1]2 (especially

119909 le 119906 le 119910 and 119909 le V le 119910 hold) Notice that 119906 ≰

V ≰ 119906 because both 1199010(119906) = 025 le 075 = 119901

0(V) and

1199011(119906) = 075 ge 025 = 119901

1(V) hold The order relation

le on [0 1]2 can be illustrated in at least two ways (seeFigure 1)

(2) When 119898 gt 2 the order relation le on [0 1]119898 can beillustrated in at least one way (see Figure 2 for the case119898 = 4 where 119909 le 119906 le 119910 119909 le V le 119910)

The Scientific World Journal 3

2 Main Results

In this section we will prove that a bipolar fuzzy set is justa very specific 119871-set that is [0 1]2-set We also put forward(or highlight) the notion of119898-polar fuzzy set (which is still aspecial 119871-set ie [0 1]119898-set although it is a generalization ofbipolar fuzzy set) and point out that many concepts whichhave been defined based on bipolar fuzzy sets and resultsrelated to these concepts can be generalized to the case of119898-polar fuzzy sets

Definition 4 An 119898-polar fuzzy set (or a [0 1]119898-set) on 119883 isexactly a mapping 119860 119883 rarr [0 1]

119898 The set of all 119898-polarfuzzy sets on119883 is denoted by119898(119883)

The following theorem shows that bipolar fuzzy sets and2-polar fuzzy sets are cryptomorphic mathematical notionsand that we can obtain concisely one from the correspondingone

Theorem 5 Let119883 be a set For each bipolar fuzzy set (120583+ 120583minus)on 119883 define a 2-polar fuzzy set

120593 (120583+ 120583minus) = 119860

120583 119883 997888rarr [0 1]

2 (1)

on 119883 by putting

119860120583(119909) = ⟨120583

+(119909) minus120583

minus(119909)⟩ (forall119909 isin 119883) (2)

Then we obtain a one-to-one correspondence

120593 119861119865 (119883) 997888rarr 2 (119883) (3)

its inverse mapping 120595 2(119883) rarr 119861119865(119883) is given by 120595(119860) =(120583+

119860 120583minus

119860) (forall119860 isin 2(119883)) 120583+

119860(119909) = 119901

0∘ 119860(119909) (forall119909 isin 119883) and

120583minus

119860(119909) = minus119901

1∘ 119860(119909) (forall119909 isin 119883)

Proof Obviously both 120593 and 120595 are mappings For each(120583+ 120583minus) isin BF(119883)

[120595 ∘ 120593 (120583+ 120583minus)] (119909)

= ⟨1199010∘ 120593 (120583

+ 120583minus) (119909) minus119901

1∘ 120593 (120583

+ 120583minus) (119909)⟩

= ⟨1199010(⟨120583+(119909) 120583

minus(119909)⟩) minus119901

1(⟨120583+(119909) 120583

minus(119909)⟩)⟩

= ⟨120583+(119909) minus120583

minus(119909)⟩ = (120583

+ 120583minus) (119909) (forall119909 isin 119883)

(4)

which means [120595 ∘ 120593(120583+ 120583minus)] = (120583+ 120583minus) Again for each 119860 isin

2(119883) and each 119909 isin 119883

[120593 ∘ 120595 (119860)] (119909) = 120593 (120583+

119860 120583minus

119860) (119909)

= ⟨120583+

119860(119909) minus120583

minus

119860(119909)⟩ = 119860 (119909)

(5)

which means 120593 ∘ 120595(119860) = 119860

Example 6 Let (120583+ 120583minus) be a bipolar fuzzy set where 119883 =

119906 V 119908 119909 119910 119911 is a six-element set and 120583+ 119883 rarr [0 1] and120583minus 119883 rarr [minus1 0] are defined by

120583+=

04

11990605

V03

1199081

1199091

11991006

119911

120583minus=

minus03

119906minus06

Vminus1

119908minus02

119909minus1

119910minus05

119911

(6)

Then the corresponding 2-polar fuzzy set on119883 is

119860120583=

⟨04 03⟩

119906⟨05 06⟩

V⟨03 1⟩

119908⟨1 02⟩

119909

⟨1 1⟩

119910⟨06 05⟩

119911

(7)

In the rest of this note we investigate the possibleapplications of 119898-polar fuzzy sets First we consider thetheoretic applications of 119898-polar fuzzy sets More preciselywe will give some remarks to illustrate how many conceptswhich have been defined based on bipolar fuzzy sets andresults related to these concepts can be generalized to the caseof119898-polar fuzzy sets (see the following Remarks 7 and 8)

Remark 7 The notions of bipolar fuzzy graph (see [4 45])and fuzzy graph (see [46 47]) can be generalized to theconvenient (because it allows a computing in computers) andintuitive notion of 119898-polar fuzzy graph An 119898-polar fuzzygraph with an underlying pair (119881 119864) (where 119864 sube 119881 times 119881

is symmetric ie it satisfies ⟨119909 119910⟩ isin 119864 hArr ⟨119910 119909⟩ isin 119864)is defined to be a pair 119866 = (119860 119861) where 119860 119881 rarr

[0 1]119898 (ie an 119898-polar fuzzy set on 119881) and 119861 119864 rarr

[0 1]119898 (ie an 119898-polar fuzzy set on 119864) satisfy 119861(⟨119909 119910⟩) le

inf119860(119909) 119860(119910) (forall⟨119909 119910⟩ isin 119864) 119860 is called the 119898-polar fuzzyvertex set of 119881 and 119861 is called the 119898-polar fuzzy edge set of119864 An 119898-polar fuzzy graph 119866 = (119860 119861) with an underlyingpair (119881 119864) and satisfying 119861(⟨119909 119910⟩) = 119861(⟨119910 119909⟩) (forall⟨119909 119910⟩ isin 119864)and 119861(⟨119909 119909⟩) = 0 (forall119909 isin 119881) is called a simple 119898-polar fuzzygraph where 0 is the smallest element of [0 1]119898 An119898-polarfuzzy graph 119866 = (119860 119861) with an underlying pair (119881 119864) andsatisfying 119861(⟨119909 119910⟩) = inf119860(119909) 119860(119910) (forall⟨119909 119910⟩ isin 119864) is calleda strong119898-polar fuzzy graphThe complement of a strong119898-polar fuzzy graph 119866 = (119860 119861) (which has an underlying pair(119881 119864)) is a strong 119898-polar fuzzy graph 119866 = (119860 119861) with anunderlying pair (119881 119864) where 119861 119864 rarr [0 1]

119898 is defined by(⟨119909 119910⟩ isin 119864 119894 isin 119898)

119901119894∘ 119861 (⟨119909 119910⟩)

= 0 119901

119894∘ 119861 (⟨119909 119910⟩) gt 0

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) 119901

119894∘ 119861 (⟨119909 119910⟩) = 0

(8)

Give two 119898-polar fuzzy graphs (with underlying pairs(1198811 1198641) and (119881

2 1198642) resp) 119866

1= (1198601 1198611) and 119866

2= (1198602 1198612)

A homomorphism from 1198661to 1198662is a mapping 119891 119881

1rarr 1198812

which satisfies1198601(119909) le 119860

2(119891(119909)) (forall119909 isin 119881

1) and119861

1(⟨119909 119910⟩) le

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864

1) An isomorphism from

1198661to 1198662is a bijective mapping 119891 119881

1rarr 119881

2which


satisfies 1198601(119909) = 119860

2(119891(119909)) (forall119909 isin 119881

1) and 119861

1(⟨119909 119910⟩) =

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864

1) A weak isomorphism from


1rarr 119881

2which is a

homomorphism and satisfies 1198601(119909) = 119860

2(119891(119909)) (forall119909 isin 119881

1)

A strong119898-polar fuzzy graph119866 is called self-complementaryif 119866 ≃ 119866 (ie there exists an isomorphism between 119866 and itscomplement 119866)

It is not difficult to verify the following conclusions (someof which generalize the corresponding results in [1 45])

(1) In a self-complementary strong 119898-polar fuzzy graph119866 = (119860 119861) (with an underlying pair (119881 119864)) we have

119901119894∘ 119861 (⟨119909 119910⟩)

= inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) minus 119901

119894∘ 119861 (⟨119909 119910⟩)

(119894 isin 119898 ⟨119909 119910⟩ isin 119864)

sum119909 = 119910

119901119894∘ 119861 (⟨119909 119910⟩)

=1

2sum119909 = 119910

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) (119894 isin 119898)

(9)

(2) A strong 119898-polar fuzzy graph 119866 = (119860 119861) (with anunderlying pair (119881 119864)) is self-complementary if andonly if it satisfies

119901119894∘ 119861 (⟨119909 119910⟩) =

1

2inf 119901

119894∘ 119860 (119909) 119901

119894∘ 119860 (119910)

(forall119894 isin 119898 forall ⟨119909 119910⟩ isin 119864)

(10)

(3) If 1198661and 119866

2are strong 119898-polar fuzzy graphs then

1198661≃ 1198662if and only if 119866

1≃ 1198662

(4) Let1198661and119866

2be strong119898-polar fuzzy graphs If there

is a weak isomorphism from 1198661to 1198662 then there is a

weak isomorphism from 1198662to 1198661

Remark 8 The fuzzifications or bipolar fuzzifications ofsome algebraic concepts (such as group 119870-algebra inclinealgebra (cf [48]) ideal filter and finite state machine) canbe generalized to the case of 119898-polar fuzzy sets An 119898-polar fuzzy set 119860 119866 rarr [0 1]

119898 is called an 119898-polarfuzzy subgroup of a group (119866 ∘) if it satisfies 119860(119909 ∘ 119910minus1) geinf119860(119909) 119860(119910) (forall119909 119910 isin 119866) An 119898-polar fuzzy set 119860 119866 rarr

[0 1]119898 is called an 119898-polar fuzzy subalgebra of a 119870-algebra

(119866 ∘ 119890 ⊙) if it satisfies 119860(119909 ∘ 119910) ge inf119860(119909) 119860(119910) (forall119909 119910 isin

119866) An 119898-polar fuzzy set 119860 119883 rarr [0 1]119898 is called an

119898-polar fuzzy subincline of an incline (119883 + lowast) if it satisfies(119909 lowast 119910) ge inf119860(119909) 119860(119910) (forall119909 119910 isin 119866) it is called an 119898-polar fuzzy ideal (resp an 119898-polar fuzzy filter) of (119883 + lowast)if it is an 119898-polar fuzzy subincline of (119883 + lowast) and satisfies119860(119909) ge 119860(119910) whenever 119909 le 119910 (resp satisfies 119860(119909) le 119860(119910)

whenever 119909 le 119910) An 119898-polar fuzzy finite state machine is atriple 119872 = (119876119883 119860) where 119876 and 119883 are finite nonemptysets (called the set of states and the set of input symbolsresp) and 119860 119876 times 119883 times 119876 rarr [0 1]

119898 is any 119898-polar fuzzy

set on119876times119883times119876 Moreover if 119861 119876 rarr [0 1]119898 is an119898-polar

fuzzy set on 119876 satisfying

119861 (119902) ge inf 119861 (119901) 119860 (⟨119901 119909 119902⟩)

(forall ⟨119901 119909 119902⟩ isin 119876 times 119883 times 119876) (11)

then1198720= (119876119883 119860 119861) is called an119898-polar subsystem of119872

Furthermore let119883lowast be the set of all words of elements of119883 offinite length and 120582 be the empty word in 119883lowast (cf [28]) Thenone can define a119898-polar fuzzy set119860lowast 119876times119883lowasttimes119876 rarr [0 1]

119898

on 119876 times 119883lowasttimes 119876 by putting

119860lowast(⟨119902 120582 119901⟩) =

1 if 119902 = 1199010 if 119902 = 119901

119860lowast(⟨119902 x 119909 119901⟩) = sup

119903isin119876

119860lowast(⟨119902 119909 119903⟩) 119860

lowast(⟨119903 119909 119901⟩)

(forall ⟨119902 x 119909 119901⟩ isin 119876 times 119883lowasttimes 119883 times 119876)

(12)

where 1 is the biggest element of [0 1]119898The following conclusions hold

(1) An 119898-polar fuzzy set 119860 119866 rarr [0 1]119898 is an 119898-polar

fuzzy subgroup of a group (119866 ∘) if and only if 119860[119886]

=

119909 isin 119866 | 119860(119909) ge 119886 is 0 or 119860[119886]

is a subgroup of(119866 ∘) (forall119886 isin [0 1]

119898)


fuzzy subalgebra of a119870-algebra (119866 ∘ 119890 ⊙) if and onlyif119860[119886]= 119909 isin 119866 | 119860(119909) ge 119886 is 0or119860

[119886]is a subalgebra

of (119866 ∘ 119890 ⊙) (forall119886 isin [0 1]119898)

(3) An 119898-polar fuzzy set 119860 119883 rarr [0 1]119898 is an 119898-

polar fuzzy subincline (resp an 119898-polar fuzzy idealan 119898-polar fuzzy filter) of an incline (119883 + lowast) if andonly if 119860

[119886]is a subincline (resp ideal filter) of

(119883 + lowast) (forall119886 isin [0 1]119898)

(4) Let 119872 = (119876119883 119860) be an 119898-polar fuzzy finitestate machine and 119861 119876 rarr [0 1]

119898 be an119898-polar fuzzy set on 119876 Then (119876119883 119860 119861) is an119898-polar subsystem of 119872 if and only if 119861(119902) ge

inf119861(119901) 119860lowast(⟨119901 x 119902⟩) (forall⟨119901 x 119902⟩ isin 119876 times 119883lowasttimes 119876)

Please see [49 50] for more results

Next we consider the applications of119898-polar fuzzy sets inreal world problems

Example 9 Let 119883 be a set consisting of five patients 119909119910 119911 119906 and V (thus 119883 = 119909 119910 119911 119906 V) They have diag-nosis data consisting of three aspects diagnosis datum of119909 is (119909) = ⟨049 046 051⟩ where datum 05 representsldquonormalrdquo or ldquoOKrdquo Suppose 119860(119910) = ⟨045 042 059⟩119860(119911) = ⟨050 040 054⟩ 119860(119906) = ⟨040 049 060⟩ and119860(V) = ⟨051 052 050⟩ Then we obtain a 3-polar fuzzy


set 119860 119883 rarr [0 1]3 which can describe the situation this

3-polar fuzzy set can also be written as follows

119860 =⟨049 046 051⟩

119909⟨045 042 059⟩

119910⟨050 040 054⟩

119911

⟨040 049 060⟩

119906⟨051 052 050⟩

V

(13)

Example 10 119898-polar fuzzy sets can be used in decisionmaking In many decision making situations it is necessaryto gather the group consensus This happens when a groupof friends decides which movie to watch when a companydecides which product design to manufacture and when ademocratic country elects its leaders For instance we con-sider here only the case of election Let 119883 = 119909 119910 119911 119906 Vbe the set of voters and 119862 = 119888

1 1198882 1198883 1198884 be the set of all

the four candidates Suppose the voting is weighted For eachcandidate 119888 isin 119862 a voter in 119909 119910 119911 can send a value in [0 1] to119888 but a voter in119883minus119909 119910 119911 can only send a value in [01 08]to 119888 Suppose 119860(119909) = ⟨09 04 001 01⟩ (which means thepreference degrees of 119909 corresponding to 119888

1 1198882 1198883 and 119888

4

are 09 04 001 and 01 resp) 119860(119910) = ⟨02 03 08 01⟩119860(119911) = ⟨08 09 08 02⟩ 119860(119906) = ⟨06 08 08 01⟩ and119860(V) = ⟨07 08 04 02⟩ Then we obtain a 4-polar fuzzy set119860 119883 rarr [0 1]

4 which can describe the situation this 4-polar fuzzy set can also be written as follows

119860 = ⟨09 04 001 01⟩

119909⟨02 03 08 01⟩

119910

⟨08 09 08 02⟩

119911

⟨06 08 08 01⟩

119906

⟨07 08 04 02⟩

V

(14)

Example 11 119898-polar fuzzy sets can be used in cooperativegames (cf [51]) Let119883 = 119909

1 1199092 119909

119899 be the set of 119899 agents

or players (119899 ge 1) 119898 = 0 1 119898 minus 1 be the set of thegrand coalitions and 119860 119883 rarr [0 1]

119898 be an 119898-polar fuzzyset where 119901

119894∘ 119860(119909) is the degree of player 119909 participating in

coalition 119894 (119909 isin 119883 119894 isin 119898) Again let V [0 1]119898 rarr 119877 (the set ofall real numbers) be a mapping satisfying V(0) = 0 Then themapping V ∘ 119860 119883 rarr 119877 is called a cooperative game whereV ∘ 119860(119909) represents the amount of money obtained by player119909 under the coalition participating ability 119860(119909) (119909 isin 119883)

(1) (a public good game compare with [51 Example 65])Suppose 119899 agents 119909

1 1199092 119909

119899want to create a facility for

joint use The cost of the facility depends on the sum ofthe participation levels (or degrees) of the agents and it isdescribed by

119896(

119899

sum119894=1

119861 (119909119894)) (15)

where 119896 [0 119899] rarr 119877 is a continuous monotonic increasingfunction with 119896(0) = 0 and 119861 119883 rarr [0 1] is a mapping Let

119860 119883 = 1199091 1199092 119909

119899 rarr [0 1]

119899 be a mapping satisfying119901119895∘ 119860(119909119894) = 119861(119909

119894) (if 119895 = 119894) or 0 (otherwise) (119894 = 1 2 119899)

Then a cooperative gamemodel V ∘119860 119883 rarr 119877 is establishedwhere V [0 1]119899 rarr 119877 is defined by

V (⟨1199041 1199042 119904

119899⟩) =

119899

sum119894=1

119892119894(119904119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894))

(forall ⟨1199041 1199042 119904

119899⟩ isin [0 1]

119899)

(16)

and the function 119892119894 [0 1] rarr 119877 is continuously monotonic

increasing with 119892119894(0) = 0 (119894 = 1 2 119899) Obviously the gain

of agent 119909119894(with participation level 119861(119909

119894)) is

V ∘ 119860 (119909119894) = 119892119894∘ 119861 (119909

119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894)) (17)

and the total gain is119899

sum119894=1

V ∘ 119860 (119909119894) =

119899

sum119894=1

119892119894∘ 119861 (119909

119894) minus 119896(

119899

sum119894=1

119861 (119909119894)) (18)

(2) There are two goods denoted 1198921and 119892

2 and three

agents 119886 119887 and 119888 with endowments (120576 120576) (1 minus 120576 0) and(0 1 minus 120576) (0 lt 120576 le 1) Let V [0 1]2 rarr 119877 be any mappingsatisfying V(⟨0 0⟩) = 0 Then the corresponding cooperativegame model is V ∘ 119860 119883 = 119886 119887 119888 rarr 119877 where

119860 = ⟨120576 120576⟩

119886⟨1 minus 120576 0⟩

119887⟨0 1 minus 120576⟩

119888

V ∘ 119860 = V (⟨120576 120576⟩)

119886V (⟨1 minus 120576 0⟩)

119887V (⟨0 1 minus 120576⟩)

119888

(19)

Example 12 119898-polar fuzzy sets can be used to defineweighted games A weighted game is a 4-tuple (119883P119882 Δ)where 119883 = 119909

1 1199092 119909

119899 is the set of 119899 players or voters

(119899 ge 2)P is a collection of fuzzy sets on119883 (called coalitions)such that (P le) is upper set (ie a fuzzy set119876 on119883 belongs toP if119876 ge 119875 for some 119875 isin P)119882 119883 rarr [0 1]

119898 is an119898-polarfuzzy set on119883 (called votingweights) andΔ sube [0 +infin)

119898minus0

(called quotas) Imagine a situation three people 119909 119910 and119911 vote for a proposal on releasing of a student Suppose that119909 casts 200 US Dollars and lose 80 hairs on her head voteseach 119910 casts 60000 US Dollars and 100 grams Cordycepssinensis votes each 119911 casts 100000 US Dollars and 100 gramsgold votes each Then an associated weighted game model is(119883P119882 Δ) where 119883 = 119909 119910 119911 and P is a collection offuzzy sets on119883 with (P le) an upper set119898 = 4

119882 = ⟨200160200 8080 0 0⟩

119909

⟨60000160200 0 100100 0⟩

119910

⟨100000160200 0 0 100100⟩

119911

Δ = ⟨200000 0 0 0⟩ ⟨100000 300 0 0⟩

⟨0 0 500 0⟩ ⟨0 0 0 65000⟩

(20)


(1) If the situation is a little simple 119909 casts [100 300]USDollars (ie the cast is between 100US Dollars and300US Dollars where [100 300] is an interval num-ber which can be looked as a point [0 +infin)

2) voteseach 119910 casts [50000 70000]US Dollars votes each 119911

casts [90000 110000]US Dollars votes each and quota is[100000 120000] Then the corresponding weighted gamemodel is (119883P119882 [100000 120000]) whereP is a collectionof fuzzy sets on119883 with (P le) an upper set119898 = 1 and

119882 = 400320400

119909120000320400

119910200000320400

119911

(21)

(2) If the situation is more simple 119909 casts 200US Dollarsvotes each 119910 casts 60000US Dollars votes each 119911 casts100000US Dollars votes each and quota is 110000 Then thecorresponding weighted game model is (119883P119882 110000)whereP = 119909 119911 119910 119911 119909 119910 119911119898 = 1 and

119882 = 200160200

11990960000160200

119910100000160200

119911

(22)

Notice that the subset 119909 119911 sube 119883 is exactly a fuzzy set 119860

119883 rarr [0 1] on119883 defined by 119860(119909) = 119860(119911) = 1 and 119860(119910) = 0

Example 13 119898-polar fuzzy sets can be used as a model forclustering or classification Consider a set 119883 consisting of 119899students 119909

1 1199092 119909

119899(119899 ge 2) in Chinese middle school

For a student 119909 isin 119883 we use integers 1199091(resp 119909

2 119909

6)

in [0 100] to denote the average score of Mathematics (respPhysics Chemistry Biology Chinese and English) and

119860 (119909) = ⟨1199091times 001 119909

2times 001 119909

3times 001

1199094times 001 119909

5times 001 119909

6times 001⟩

(23)

Then we obtain a 6-polar fuzzy set model 119860 119883 rarr [0 1]6

which can be used for clustering or classification of thesestudents

Example 14 119898-polar fuzzy sets can be used to define multi-valued relations

(1) Consider a set 119883 consisting of 119899 net users (resppatients) 119909

1 1199092 119909

119899(119899 ge 2) For net users (resp

patients) 119909 119910 isin 119883 we use (119909 119910 119895) to denote thesimilarity between 119909 and 119910 in 119895th aspect (1 le

119895 le 119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) =

⟨(119909 119910 1) (119909 119910 2) (119909 119910119898)⟩ Then we obtain an119898-polar fuzzy set 119860 119883 rarr [0 1]

119898 which is amultivalued similarity relation

(2) Consider a set 119883 consisting of 119899 people 1199091 1199092

119909119899(119899 ge 2) in a social network For 119909 119910 isin

119883 we use (119909 119910 119895) to denote the degree of connec-tion between 119909 and 119910 in 119895th aspect (1 le 119895 le

119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) = ⟨(119909 119910 1)(119909 119910 2) (119909 119910119898)⟩ Then we obtain an 119898-polarfuzzy set 119860 119883 rarr [0 1]

119898 which is a multivaluedsocial graph (or multivalued social network) model

3 Conclusion

In this note we show that the enthusiastically studied notionof bipolar fuzzy set is actually a synonym of a [0 1]2-set (wecall it 2-polar fuzzy set) and thus we highlight the notion of119898-polar fuzzy set (actually a [0 1]119898-set119898 ge 2) The119898-polarfuzzy sets not only have real backgrounds (eg ldquomultipolarinformationrdquo exists) but also have applications in both theoryand real world problems (which have been illustrated byexamples)

Conflict of Interests

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

Acknowledgments

This work was supported by the International Science andTechnology Cooperation Foundation of China (Grant no2012DFA11270) and the National Natural Science Foundationof China (Grant no 11071151)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] U Hohle and S E Rodabaugh EdsMathematics of Fuzzy SetsLogic Topology and Measure Theory The Handbooks of FuzzySets Series Kluwer Academic Dordrecht The Netherlands1999

[3] W R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic and FuzzyInformation Processing Society Biannual Conference pp 305ndash309 San Antonio Tex USA December 1994

[4] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[5] MAkram ldquoBipolar fuzzy graphswith applicationsrdquoKnowledge-Based Systems vol 39 pp 1ndash8 2013

[6] M Akram W Chen and Y Yin ldquoBipolar fuzzy Lie superalge-brasrdquo Quasigroups and Related Systems vol 20 no 2 pp 139ndash156 2012

[7] M Akram S G Li and K P Shum ldquoAntipodal bipolar fuzzygraphsrdquo Italian Journal of Pure and Applied Mathematics vol31 pp 425ndash438 2013

[8] M Akram A B Saeid K P Shum and B L Meng ldquoBipolarfuzzy K-algebrasrdquo International Journal of Fuzzy Systems vol12 no 3 pp 252ndash259 2010

[9] L Amgoud C Cayrol M C Lagasquie-Schiex and P LivetldquoOn bipolarity in argumentation frameworksrdquo InternationalJournal of Intelligent Systems vol 23 no 10 pp 1062ndash1093 2008

[10] H Y Ban M J Kim and Y J Park ldquoBipolar fuzzy ideals withoperators in semigroupsrdquo Annals of Fuzzy Mathematics andInformatics vol 4 no 2 pp 253ndash265 2012

[11] S Benferhat D Dubois S Kaci and H Prade ldquoBipolar possi-bility theory in preferencemodeling representation fusion and


optimal solutionsrdquo Information Fusion vol 7 no 1 pp 135ndash1502006

[12] S Bhattacharya and S Roy ldquoStudy on bipolar fuzzy-roughcontrol theoryrdquo International Mathematical Forum vol 7 no41 pp 2019ndash2025 2012

[13] I Bloch ldquoDilation and erosion of spatial bipolar fuzzy setsrdquo inApplications of Fuzzy Sets Theory F Masulli S Mitra and GPasi Eds vol 4578 of Lecture Notes in Computer Science pp385ndash393 Springer Berlin Germany 2007

[14] I Bloch ldquoBipolar fuzzy spatial information geometry mor-phology spatial reasoningrdquo in Methods for Handling ImperfectSpatial Information R Jeansoulin O Papini H Prade andS Schockaert Eds vol 256 of Studies in Fuzziness and SoftComputing pp 75ndash102 Springer Berlin Germany 2010

[15] I Bloch ldquoLattices of fuzzy sets and bipolar fuzzy sets andmath-ematical morphologyrdquo Information Sciences vol 181 no 10 pp2002ndash2015 2011

[16] I Bloch ldquoMathematical morphology on bipolar fuzzy sets gen-eral algebraic frameworkrdquo International Journal of ApproximateReasoning vol 53 no 7 pp 1031ndash1060 2012

[17] I Bloch and J Atif ldquoDistance to bipolar information frommorphological dilationrdquo in Proceedings of the 8th Conference ofthe European Society for Fuzzy Logic and Technology pp 266ndash273 2013

[18] J F Bonnefon ldquoTwo routes for bipolar information processingand a blind spot in betweenrdquo International Journal of IntelligentSystems vol 23 no 9 pp 923ndash929 2008

[19] P Bosc and O Pivert ldquoOn a fuzzy bipolar relational algebrardquoInformation Sciences vol 219 pp 1ndash16 2013

[20] D Dubois S Kaci and H Prade ldquoBipolarity in reasoningand decision an introductionrdquo in Proceedings of the Interna-tional Conference on Information Processing andManagement ofUncertainty pp 959ndash966 2004

[21] D Dubois and H Prade ldquoAn overview of the asymmetricbipolar representation of positive and negative information inpossibility theoryrdquo Fuzzy Sets and Systems vol 160 no 10 pp1355ndash1366 2009

[22] U Dudziak and B Pekala ldquoEquivalent bipolar fuzzy relationsrdquoFuzzy Sets and Systems vol 161 no 2 pp 234ndash253 2010

[23] H Fargier and N Wilson ldquoAlgebraic structures for bipo-lar constraint-based reasoningrdquo in Symbolic and QuantitativeApproaches to Reasoning with Uncertainty vol 4724 of LectureNotes in Computer Science pp 623ndash634 Springer BerlinGermany 2007

[24] M Grabisch S Greco and M Pirlot ldquoBipolar and bivari-ate models in multicriteria decision analysis descriptive andconstructive approachesrdquo International Journal of IntelligentSystems vol 23 no 9 pp 930ndash969 2008

[25] M M Hasankhani and A B Saeid ldquoHyper MV-algebrasdefined by bipolar-valued fuzzy setsrdquo Annals of West Universityof Timisoara-Mathematics vol 50 no 1 pp 39ndash50 2012

[26] C Hudelot J Atif and I Bloch ldquoIntegrating bipolar fuzzymathematical morphology in description logics for spatialreasoningrdquo Frontiers in Artificial Intelligence and Applicationsvol 215 pp 497ndash502 2010

[27] Y B Jun M S Kang andH S Kim ldquoBipolar fuzzy hyper BCK-ideals in hyper BCK-algebrasrdquo Iranian Journal of Fuzzy Systemsvol 8 no 2 pp 105ndash120 2011

[28] Y B Jun and J Kavikumar ldquoBipolar fuzzy finite state machinesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 34no 1 pp 181ndash188 2011

[29] Y B Jun H S Kim and K J Lee ldquoBipolar fuzzy translationin BCKBCI-algebrardquo Journal of the ChungcheongMathematicalSociety vol 22 no 3 pp 399ndash408 2009

[30] Y B Jun and C H Park ldquoFilters of BCH-algebras based onbipolar-valued fuzzy setsrdquo International Mathematical Forumvol 4 no 13 pp 631ndash643 2009

[31] S Kaci ldquoLogical formalisms for representing bipolar prefer-encesrdquo International Journal of Intelligent Systems vol 23 no9 pp 985ndash997 2008

[32] K J Lee ldquoBipolar fuzzy subalgebras and bipolar fuzzy idealsof BCKBCI-algebrasrdquo Bulletin of the Malaysian MathematicalSciences Society vol 32 no 3 pp 361ndash373 2009

[33] K J Lee and Y B Jun ldquoBipolar fuzzy a-ideals of BCI-algebrasrdquoCommunications of the KoreanMathematical Society vol 26 no4 pp 531ndash542 2011

[34] KM Lee ldquoComparison of interval-valued fuzzy sets intuition-istic fuzzy sets and bipolar-valued fuzzy setsrdquo Journal of FuzzyLogic Intelligent Systems vol 14 no 2 pp 125ndash129 2004

[35] R Muthuraj and M Sridharan ldquoBipolar anti fuzzy HX groupand its lower level sub HX groupsrdquo Journal of Physical Sciencesvol 16 pp 157ndash169 2012

[36] S Narayanamoorthy and A Tamilselvi ldquoBipolar fuzzy linegraph of a bipolar fuzzy hypergraphrdquo Cybernetics and Informa-tion Technologies vol 13 no 1 pp 13ndash17 2013

[37] E Raufaste and S Vautier ldquoAn evolutionist approach toinformation bipolarity representations and affects in humancognitionrdquo International Journal of Intelligent Systems vol 23no 8 pp 878ndash897 2008

[38] A B Saeid ldquoBM-algebras defined by bipolar-valued setsrdquoIndian Journal of Science and Technology vol 5 no 2 pp 2071ndash2078 2012

[39] S Samanta and M Pal ldquoIrregular bipolar fuzzy graphsrdquo Inter-national Journal of Applications of Fuzzy Sets vol 2 no 2 pp91ndash102 2012

[40] H L Yang S G Li Z L Guo andCHMa ldquoTransformation ofbipolar fuzzy rough set modelsrdquo Knowledge-Based Systems vol27 pp 60ndash68 2012

[41] H L Yang S G Li S Y Wang and J Wang ldquoBipolar fuzzyrough set model on two different universes and its applicationrdquoKnowledge-Based Systems vol 35 pp 94ndash101 2012

[42] W R Zhang ldquoEquilibrium relations and bipolar fuzzy cluster-ingrdquo in Proceedings of the 18th International Conference of theNorth American Fuzzy Information Processing Society (NAFIPS99) pp 361ndash365 June 1999

[43] W R Zhang Ed YinYang Bipolar Relativity A UnifyingTheoryof Nature Agents and Causality with Applications in QuantumComputing Cognitive Informatics and Life Sciences IGI Global2011

[44] W R Zhang ldquoBipolar quantum logic gates and quantumcellular combinatoricsmdasha logical extension to quantum entan-glementrdquo Journal of Quantum Information Science vol 3 no 2pp 93ndash105 2013

[45] H L Yang S G Li W H Yang and Y Lu ldquoNotes on lsquobipolarfuzzy graphsrsquordquo Information Sciences vol 242 pp 113ndash121 2013

[46] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions to Cognitive and Decision Process L A Zadeh K S Fuand M Shimura Eds pp 77ndash95 Academic Press New YorkNY USA 1975

[47] R T Yeh and S Y Bang ldquoFuzzy relations fuzzy graphs and theirapplication to clustering analysisrdquo in Fuzzy Sets andTheir Appli-cations to Cognitive and Decision Process L A Zadeh K S Fu


and M Shimura Eds pp 338ndash353 Academic Press New YorkNY USA 1975

[48] Z Q Cao K H Kim and F W Roush Incline Algebra andApplications Ellis Horwood Series in Mathematics and ItsApplicationsHalstedPress ChichesterUK JohnWileyampSonsNew York NY USA 1984

[49] Y M Li ldquoFinite automata theory with membership values inlatticesrdquo Information Sciences vol 181 no 5 pp 1003ndash1017 2011

[50] J H Jin Q G Li and Y M Li ldquoAlgebraic properties of L-fuzzy finite automatardquo Information Sciences vol 234 pp 182ndash202 2013

[51] L J Xie and M Grabisch ldquoThe core of bicapacities and bipolargamesrdquo Fuzzy Sets and Systems vol 158 no 9 pp 1000ndash10122007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


ldquoyangrdquo are the two sides Yin is the feminine or negative sideof a system and yang is the masculine or positive side of asystem The coexistence equilibrium and harmony of thetwo sides are considered a key for the mental and physicalhealth of a person as well as for the stability and prosperity ofa social systemThus bipolar fuzzy sets indeed have potentialimpacts on many fields including artificial intelligencecomputer science information science cognitive sciencedecision science management science economics neuralscience quantum computing medical science and socialscience (cf [4ndash45]) In recent years bipolar fuzzy sets seemto have been studied and applied a bit enthusiastically and abit increasingly (cf [4ndash45]) This is the chief motivation forus to introduce and study119898-polar fuzzy sets

The first object of this note is to answer the followingquestion on bipolar fuzzy sets

Question 1 Is bipolar fuzzy set a very intuitive 119871-setThe answer to Question 1 is positive We will prove in

this note that there is a natural one-to-one correspondencebetween BF(119883) and 2(119883) (for the set of all [0 1]2-sets on119883 see Theorem 5) which preserves all involved propertiesThis makes the notion of bipolar fuzzy set more intuitiveSince properties of 119871-sets have already been studied verydeeply and exhaustively this one-to-one correspondencemay be beneficial for both researchers interested in above-mentioned papers and related fields (because they can usethese properties directly and even cooperate with theoreticalfuzzy mathematicians for a possible higher-level research)and theoretical fuzzy mathematicians as well (because coop-eration with applied fuzzy mathematicians and practitionersprobably makes their research more useful)

We notice that ldquomultipolar informationrdquo (not just bipolarinformation which corresponds to two-valued logic) existsbecause data for a real world problem are sometimes from119899 agents (119899 ge 2) For example the exact degree oftelecommunication safety ofmankind is a point in [0 1]119899 (119899 asymp7times109) because different person has beenmonitored different

timesThere aremany other examples truth degrees of a logicformula which are based on 119899 logic implication operators(119899 ge 2) similarity degrees of two logic formulas which arebased on 119899 logic implication operators (119899 ge 2) orderingresults of a magazine ordering results of a university andinclusion degrees (resp accuracy measures roughmeasuresapproximation qualities fuzziness measures and decisionperformance evaluations) of a rough set Thus our secondobject of this note is to answer the following question onextensions of bipolar fuzzy sets

Question 2 How to generalize bipolar fuzzy sets to multipo-lar fuzzy sets and how to generalize results on bipolar fuzzysets to the case of multipolar fuzzy sets

The idea to answer Question 2 is from the answer toQuestion 1 intuitiveness of the point-wise order on [0 1]

119898

(see Remark 3) and the proven corresponding results onbipolar fuzzy sets We put forward the notion of 119898-polarfuzzy set (an extension of bipolar fuzzy set) and point outthatmany concepts which have been defined based on bipolarfuzzy sets and many results related to these concepts can be

y

x

u

vu

uv

y y

v

x x

Figure 1

y y

v

vu

u

x x

Figure 2

generalized to the case of 119898-polar fuzzy sets (see Remarks 7and 8 for details)

Apart from the backgrounds (eg ldquomultipolar informa-tionrdquo) of119898-polar fuzzy sets the following question on furtherapplications (particularly further applications in real worldproblems) of119898-polar fuzzy sets should also be considered

Question 3 How to find further possible applications of 119898-polar fuzzy sets in real world problems

Question 3 can be answered as in the case of bipolar fuzzysets since researches or modelings on real world problemsoften involve multiagent multiattribute multiobject multi-index multipolar information uncertainty orand limitsprocess We will give examples to demonstrate it (see Exam-ples 9ndash14)

Remark 3 In this note [0 1]119898 (119898-power of [0 1]) is con-sidered a poset with the point-wise order le where 119898 is anarbitrary ordinal number (wemake an appointment that119898 =

119899 | 119899 lt 119898 when 119898 gt 0) le (which is actually very intuitiveas illustrated below) is defined by 119909 le 119910 hArr 119901

119894(119909) le 119901

119894(119910) for

each 119894 isin 119898 (119909 119910 isin [0 1]119898) and 119901

119894 [0 1]

119898rarr [0 1] is the

119894th projection mapping (119894 isin 119898)

(1) When119898 = 2 [0 1]2 is the ordinary closed unit squarein Euclidean plane 1198772 The righter (resp the upper) apoint in this square is the larger it is Let 119909 = ⟨0 0⟩ =0 (the smallest element of [0 1]2) 119906 = ⟨025 075⟩V = ⟨075 025⟩ and 119910 = ⟨1 1⟩ (the largest element of[0 1]2) Then 119909 le 119911 le 119910 for all 119911 isin [0 1]2 (especially

119909 le 119906 le 119910 and 119909 le V le 119910 hold) Notice that 119906 ≰

V ≰ 119906 because both 1199010(119906) = 025 le 075 = 119901

0(V) and

1199011(119906) = 075 ge 025 = 119901

1(V) hold The order relation

le on [0 1]2 can be illustrated in at least two ways (seeFigure 1)

(2) When 119898 gt 2 the order relation le on [0 1]119898 can beillustrated in at least one way (see Figure 2 for the case119898 = 4 where 119909 le 119906 le 119910 119909 le V le 119910)


2 Main Results






120593 (120583+ 120583minus) = 119860

120583 119883 997888rarr [0 1]

2 (1)


119860120583(119909) = ⟨120583

+(119909) minus120583



120593 119861119865 (119883) 997888rarr 2 (119883) (3)


119860 120583minus

119860) (forall119860 isin 2(119883)) 120583+

119860(119909) = 119901

0∘ 119860(119909) (forall119909 isin 119883) and

120583minus

119860(119909) = minus119901

1∘ 119860(119909) (forall119909 isin 119883)


[120595 ∘ 120593 (120583+ 120583minus)] (119909)

= ⟨1199010∘ 120593 (120583

+ 120583minus) (119909) minus119901

1∘ 120593 (120583

+ 120583minus) (119909)⟩

= ⟨1199010(⟨120583+(119909) 120583

minus(119909)⟩) minus119901

1(⟨120583+(119909) 120583

minus(119909)⟩)⟩

= ⟨120583+(119909) minus120583

minus(119909)⟩ = (120583


(4)


2(119883) and each 119909 isin 119883

[120593 ∘ 120595 (119860)] (119909) = 120593 (120583+

119860 120583minus

119860) (119909)

= ⟨120583+

119860(119909) minus120583

minus

119860(119909)⟩ = 119860 (119909)

(5)

which means 120593 ∘ 120595(119860) = 119860



120583+=

04

11990605

V03

1199081

1199091

11991006

119911

120583minus=

minus03

119906minus06

Vminus1

119908minus02

119909minus1

119910minus05

119911

(6)


119860120583=

⟨04 03⟩

119906⟨05 06⟩

V⟨03 1⟩

119908⟨1 02⟩

119909

⟨1 1⟩

119910⟨06 05⟩

119911

(7)








119901119894∘ 119861 (⟨119909 119910⟩)

= 0 119901

119894∘ 119861 (⟨119909 119910⟩) gt 0

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) 119901

119894∘ 119861 (⟨119909 119910⟩) = 0

(8)


2 1198642) resp) 119866

1= (1198601 1198611) and 119866

2= (1198602 1198612)


1rarr 1198812


2(119891(119909)) (forall119909 isin 119881

1) and119861

1(⟨119909 119910⟩) le

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864



1rarr 119881

2which


satisfies 1198601(119909) = 119860

2(119891(119909)) (forall119909 isin 119881

1) and 119861

1(⟨119909 119910⟩) =

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864



1rarr 119881

2which is a


2(119891(119909)) (forall119909 isin 119881

1)




119901119894∘ 119861 (⟨119909 119910⟩)

= inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) minus 119901

119894∘ 119861 (⟨119909 119910⟩)

(119894 isin 119898 ⟨119909 119910⟩ isin 119864)

sum119909 = 119910

119901119894∘ 119861 (⟨119909 119910⟩)

=1

2sum119909 = 119910

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) (119894 isin 119898)

(9)


119901119894∘ 119861 (⟨119909 119910⟩) =

1

2inf 119901

119894∘ 119860 (119909) 119901

119894∘ 119860 (119910)


(10)

(3) If 1198661and 119866


1198661≃ 1198662if and only if 119866

1≃ 1198662

(4) Let1198661and119866














119861 (119902) ge inf 119861 (119901) 119860 (⟨119901 119909 119902⟩)




119898


119860lowast(⟨119902 120582 119901⟩) =

1 if 119902 = 1199010 if 119902 = 119901

119860lowast(⟨119902 x 119909 119901⟩) = sup

119903isin119876

119860lowast(⟨119902 119909 119903⟩) 119860

lowast(⟨119903 119909 119901⟩)


(12)




=

119909 isin 119866 | 119860(119909) ge 119886 is 0 or 119860[119886]


119898)




of (119866 ∘ 119890 ⊙) (forall119886 isin [0 1]119898)














119860 =⟨049 046 051⟩

119909⟨045 042 059⟩

119910⟨050 040 054⟩

119911

⟨040 049 060⟩

119906⟨051 052 050⟩

V

(13)


1 1198882 1198883 1198884 be the set of all


1 1198882 1198883 and 119888

4



119860 = ⟨09 04 001 01⟩

119909⟨02 03 08 01⟩

119910

⟨08 09 08 02⟩

119911

⟨06 08 08 01⟩

119906

⟨07 08 04 02⟩

V

(14)


1 1199092 119909







1 1199092 119909



119896(

119899

sum119894=1

119861 (119909119894)) (15)


119860 119883 = 1199091 1199092 119909

119899 rarr [0 1]


119894) (if 119895 = 119894) or 0 (otherwise) (119894 = 1 2 119899)


V (⟨1199041 1199042 119904

119899⟩) =

119899

sum119894=1

119892119894(119904119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894))

(forall ⟨1199041 1199042 119904

119899⟩ isin [0 1]

119899)

(16)




119894)) is

V ∘ 119860 (119909119894) = 119892119894∘ 119861 (119909

119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894)) (17)


sum119894=1

V ∘ 119860 (119909119894) =

119899

sum119894=1

119892119894∘ 119861 (119909

119894) minus 119896(

119899

sum119894=1

119861 (119909119894)) (18)


2 and three


119860 = ⟨120576 120576⟩

119886⟨1 minus 120576 0⟩

119887⟨0 1 minus 120576⟩

119888

V ∘ 119860 = V (⟨120576 120576⟩)

119886V (⟨1 minus 120576 0⟩)

119887V (⟨0 1 minus 120576⟩)

119888

(19)


1 1199092 119909




119898minus0


119882 = ⟨200160200 8080 0 0⟩

119909

⟨60000160200 0 100100 0⟩

119910

⟨100000160200 0 0 100100⟩

119911

Δ = ⟨200000 0 0 0⟩ ⟨100000 300 0 0⟩

⟨0 0 500 0⟩ ⟨0 0 0 65000⟩

(20)





119882 = 400320400

119909120000320400

119910200000320400

119911

(21)


119882 = 200160200

11990960000160200

119910100000160200

119911

(22)




1 1199092 119909



2 119909

6)


119860 (119909) = ⟨1199091times 001 119909

2times 001 119909

3times 001

1199094times 001 119909

5times 001 119909

6times 001⟩

(23)





1 1199092 119909



119895 le 119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) =








3 Conclusion




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 Main Results






120593 (120583+ 120583minus) = 119860

120583 119883 997888rarr [0 1]

2 (1)


119860120583(119909) = ⟨120583

+(119909) minus120583



120593 119861119865 (119883) 997888rarr 2 (119883) (3)


119860 120583minus

119860) (forall119860 isin 2(119883)) 120583+

119860(119909) = 119901

0∘ 119860(119909) (forall119909 isin 119883) and

120583minus

119860(119909) = minus119901

1∘ 119860(119909) (forall119909 isin 119883)


[120595 ∘ 120593 (120583+ 120583minus)] (119909)

= ⟨1199010∘ 120593 (120583

+ 120583minus) (119909) minus119901

1∘ 120593 (120583

+ 120583minus) (119909)⟩

= ⟨1199010(⟨120583+(119909) 120583

minus(119909)⟩) minus119901

1(⟨120583+(119909) 120583

minus(119909)⟩)⟩

= ⟨120583+(119909) minus120583

minus(119909)⟩ = (120583


(4)


2(119883) and each 119909 isin 119883

[120593 ∘ 120595 (119860)] (119909) = 120593 (120583+

119860 120583minus

119860) (119909)

= ⟨120583+

119860(119909) minus120583

minus

119860(119909)⟩ = 119860 (119909)

(5)

which means 120593 ∘ 120595(119860) = 119860



120583+=

04

11990605

V03

1199081

1199091

11991006

119911

120583minus=

minus03

119906minus06

Vminus1

119908minus02

119909minus1

119910minus05

119911

(6)


119860120583=

⟨04 03⟩

119906⟨05 06⟩

V⟨03 1⟩

119908⟨1 02⟩

119909

⟨1 1⟩

119910⟨06 05⟩

119911

(7)








119901119894∘ 119861 (⟨119909 119910⟩)

= 0 119901

119894∘ 119861 (⟨119909 119910⟩) gt 0

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) 119901

119894∘ 119861 (⟨119909 119910⟩) = 0

(8)


2 1198642) resp) 119866

1= (1198601 1198611) and 119866

2= (1198602 1198612)


1rarr 1198812


2(119891(119909)) (forall119909 isin 119881

1) and119861

1(⟨119909 119910⟩) le

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864



1rarr 119881

2which


satisfies 1198601(119909) = 119860

2(119891(119909)) (forall119909 isin 119881

1) and 119861

1(⟨119909 119910⟩) =

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864



1rarr 119881

2which is a


2(119891(119909)) (forall119909 isin 119881

1)




119901119894∘ 119861 (⟨119909 119910⟩)

= inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) minus 119901

119894∘ 119861 (⟨119909 119910⟩)

(119894 isin 119898 ⟨119909 119910⟩ isin 119864)

sum119909 = 119910

119901119894∘ 119861 (⟨119909 119910⟩)

=1

2sum119909 = 119910

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) (119894 isin 119898)

(9)


119901119894∘ 119861 (⟨119909 119910⟩) =

1

2inf 119901

119894∘ 119860 (119909) 119901

119894∘ 119860 (119910)


(10)

(3) If 1198661and 119866


1198661≃ 1198662if and only if 119866

1≃ 1198662

(4) Let1198661and119866














119861 (119902) ge inf 119861 (119901) 119860 (⟨119901 119909 119902⟩)




119898


119860lowast(⟨119902 120582 119901⟩) =

1 if 119902 = 1199010 if 119902 = 119901

119860lowast(⟨119902 x 119909 119901⟩) = sup

119903isin119876

119860lowast(⟨119902 119909 119903⟩) 119860

lowast(⟨119903 119909 119901⟩)


(12)




=

119909 isin 119866 | 119860(119909) ge 119886 is 0 or 119860[119886]


119898)




of (119866 ∘ 119890 ⊙) (forall119886 isin [0 1]119898)














119860 =⟨049 046 051⟩

119909⟨045 042 059⟩

119910⟨050 040 054⟩

119911

⟨040 049 060⟩

119906⟨051 052 050⟩

V

(13)


1 1198882 1198883 1198884 be the set of all


1 1198882 1198883 and 119888

4



119860 = ⟨09 04 001 01⟩

119909⟨02 03 08 01⟩

119910

⟨08 09 08 02⟩

119911

⟨06 08 08 01⟩

119906

⟨07 08 04 02⟩

V

(14)


1 1199092 119909







1 1199092 119909



119896(

119899

sum119894=1

119861 (119909119894)) (15)


119860 119883 = 1199091 1199092 119909

119899 rarr [0 1]


119894) (if 119895 = 119894) or 0 (otherwise) (119894 = 1 2 119899)


V (⟨1199041 1199042 119904

119899⟩) =

119899

sum119894=1

119892119894(119904119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894))

(forall ⟨1199041 1199042 119904

119899⟩ isin [0 1]

119899)

(16)




119894)) is

V ∘ 119860 (119909119894) = 119892119894∘ 119861 (119909

119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894)) (17)


sum119894=1

V ∘ 119860 (119909119894) =

119899

sum119894=1

119892119894∘ 119861 (119909

119894) minus 119896(

119899

sum119894=1

119861 (119909119894)) (18)


2 and three


119860 = ⟨120576 120576⟩

119886⟨1 minus 120576 0⟩

119887⟨0 1 minus 120576⟩

119888

V ∘ 119860 = V (⟨120576 120576⟩)

119886V (⟨1 minus 120576 0⟩)

119887V (⟨0 1 minus 120576⟩)

119888

(19)


1 1199092 119909




119898minus0


119882 = ⟨200160200 8080 0 0⟩

119909

⟨60000160200 0 100100 0⟩

119910

⟨100000160200 0 0 100100⟩

119911

Δ = ⟨200000 0 0 0⟩ ⟨100000 300 0 0⟩

⟨0 0 500 0⟩ ⟨0 0 0 65000⟩

(20)





119882 = 400320400

119909120000320400

119910200000320400

119911

(21)


119882 = 200160200

11990960000160200

119910100000160200

119911

(22)




1 1199092 119909



2 119909

6)


119860 (119909) = ⟨1199091times 001 119909

2times 001 119909

3times 001

1199094times 001 119909

5times 001 119909

6times 001⟩

(23)





1 1199092 119909



119895 le 119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) =








3 Conclusion




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










satisfies 1198601(119909) = 119860

2(119891(119909)) (forall119909 isin 119881

1) and 119861

1(⟨119909 119910⟩) =

1198612(⟨119891(119909) 119891(119910)⟩) (forall⟨119909 119910⟩ isin 119864



1rarr 119881

2which is a


2(119891(119909)) (forall119909 isin 119881

1)




119901119894∘ 119861 (⟨119909 119910⟩)

= inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) minus 119901

119894∘ 119861 (⟨119909 119910⟩)

(119894 isin 119898 ⟨119909 119910⟩ isin 119864)

sum119909 = 119910

119901119894∘ 119861 (⟨119909 119910⟩)

=1

2sum119909 = 119910

inf 119901119894∘ 119860 (119909) 119901

119894∘ 119860 (119910) (119894 isin 119898)

(9)


119901119894∘ 119861 (⟨119909 119910⟩) =

1

2inf 119901

119894∘ 119860 (119909) 119901

119894∘ 119860 (119910)


(10)

(3) If 1198661and 119866


1198661≃ 1198662if and only if 119866

1≃ 1198662

(4) Let1198661and119866














119861 (119902) ge inf 119861 (119901) 119860 (⟨119901 119909 119902⟩)




119898


119860lowast(⟨119902 120582 119901⟩) =

1 if 119902 = 1199010 if 119902 = 119901

119860lowast(⟨119902 x 119909 119901⟩) = sup

119903isin119876

119860lowast(⟨119902 119909 119903⟩) 119860

lowast(⟨119903 119909 119901⟩)


(12)




=

119909 isin 119866 | 119860(119909) ge 119886 is 0 or 119860[119886]


119898)




of (119866 ∘ 119890 ⊙) (forall119886 isin [0 1]119898)














119860 =⟨049 046 051⟩

119909⟨045 042 059⟩

119910⟨050 040 054⟩

119911

⟨040 049 060⟩

119906⟨051 052 050⟩

V

(13)


1 1198882 1198883 1198884 be the set of all


1 1198882 1198883 and 119888

4



119860 = ⟨09 04 001 01⟩

119909⟨02 03 08 01⟩

119910

⟨08 09 08 02⟩

119911

⟨06 08 08 01⟩

119906

⟨07 08 04 02⟩

V

(14)


1 1199092 119909







1 1199092 119909



119896(

119899

sum119894=1

119861 (119909119894)) (15)


119860 119883 = 1199091 1199092 119909

119899 rarr [0 1]


119894) (if 119895 = 119894) or 0 (otherwise) (119894 = 1 2 119899)


V (⟨1199041 1199042 119904

119899⟩) =

119899

sum119894=1

119892119894(119904119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894))

(forall ⟨1199041 1199042 119904

119899⟩ isin [0 1]

119899)

(16)




119894)) is

V ∘ 119860 (119909119894) = 119892119894∘ 119861 (119909

119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894)) (17)


sum119894=1

V ∘ 119860 (119909119894) =

119899

sum119894=1

119892119894∘ 119861 (119909

119894) minus 119896(

119899

sum119894=1

119861 (119909119894)) (18)


2 and three


119860 = ⟨120576 120576⟩

119886⟨1 minus 120576 0⟩

119887⟨0 1 minus 120576⟩

119888

V ∘ 119860 = V (⟨120576 120576⟩)

119886V (⟨1 minus 120576 0⟩)

119887V (⟨0 1 minus 120576⟩)

119888

(19)


1 1199092 119909




119898minus0


119882 = ⟨200160200 8080 0 0⟩

119909

⟨60000160200 0 100100 0⟩

119910

⟨100000160200 0 0 100100⟩

119911

Δ = ⟨200000 0 0 0⟩ ⟨100000 300 0 0⟩

⟨0 0 500 0⟩ ⟨0 0 0 65000⟩

(20)





119882 = 400320400

119909120000320400

119910200000320400

119911

(21)


119882 = 200160200

11990960000160200

119910100000160200

119911

(22)




1 1199092 119909



2 119909

6)


119860 (119909) = ⟨1199091times 001 119909

2times 001 119909

3times 001

1199094times 001 119909

5times 001 119909

6times 001⟩

(23)





1 1199092 119909



119895 le 119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) =








3 Conclusion




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119860 =⟨049 046 051⟩

119909⟨045 042 059⟩

119910⟨050 040 054⟩

119911

⟨040 049 060⟩

119906⟨051 052 050⟩

V

(13)


1 1198882 1198883 1198884 be the set of all


1 1198882 1198883 and 119888

4



119860 = ⟨09 04 001 01⟩

119909⟨02 03 08 01⟩

119910

⟨08 09 08 02⟩

119911

⟨06 08 08 01⟩

119906

⟨07 08 04 02⟩

V

(14)


1 1199092 119909







1 1199092 119909



119896(

119899

sum119894=1

119861 (119909119894)) (15)


119860 119883 = 1199091 1199092 119909

119899 rarr [0 1]


119894) (if 119895 = 119894) or 0 (otherwise) (119894 = 1 2 119899)


V (⟨1199041 1199042 119904

119899⟩) =

119899

sum119894=1

119892119894(119904119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894))

(forall ⟨1199041 1199042 119904

119899⟩ isin [0 1]

119899)

(16)




119894)) is

V ∘ 119860 (119909119894) = 119892119894∘ 119861 (119909

119894) minus

1

119899119896(

119899

sum119894=1

119861 (119909119894)) (17)


sum119894=1

V ∘ 119860 (119909119894) =

119899

sum119894=1

119892119894∘ 119861 (119909

119894) minus 119896(

119899

sum119894=1

119861 (119909119894)) (18)


2 and three


119860 = ⟨120576 120576⟩

119886⟨1 minus 120576 0⟩

119887⟨0 1 minus 120576⟩

119888

V ∘ 119860 = V (⟨120576 120576⟩)

119886V (⟨1 minus 120576 0⟩)

119887V (⟨0 1 minus 120576⟩)

119888

(19)


1 1199092 119909




119898minus0


119882 = ⟨200160200 8080 0 0⟩

119909

⟨60000160200 0 100100 0⟩

119910

⟨100000160200 0 0 100100⟩

119911

Δ = ⟨200000 0 0 0⟩ ⟨100000 300 0 0⟩

⟨0 0 500 0⟩ ⟨0 0 0 65000⟩

(20)





119882 = 400320400

119909120000320400

119910200000320400

119911

(21)


119882 = 200160200

11990960000160200

119910100000160200

119911

(22)




1 1199092 119909



2 119909

6)


119860 (119909) = ⟨1199091times 001 119909

2times 001 119909

3times 001

1199094times 001 119909

5times 001 119909

6times 001⟩

(23)





1 1199092 119909



119895 le 119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) =








3 Conclusion




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119882 = 400320400

119909120000320400

119910200000320400

119911

(21)


119882 = 200160200

11990960000160200

119910100000160200

119911

(22)




1 1199092 119909



2 119909

6)


119860 (119909) = ⟨1199091times 001 119909

2times 001 119909

3times 001

1199094times 001 119909

5times 001 119909

6times 001⟩

(23)





1 1199092 119909



119895 le 119898 119898 ge 2) and let 119860(119909 119910) = 119860(119910 119909) =








3 Conclusion




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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