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Fuzzy Sets and Systems 161 (2010) 2709–2723www.elsevier.com/locate/fss

Connectedness of refined Goetschel–Voxman fuzzy matroids�

Xiao-Nan Lia, San-yang Liua, Sheng-gang Lib,∗aDepartment of Mathematics, Xidian University, 710071 Xi’an, China

bCollege of Mathematics and Information Science, Shaanxi Normal University, 710062 Xi’an, China

Received 15 December 2008; received in revised form 8 December 2009; accepted 23 April 2010Available online 6 May 2010

Abstract

This paper studies connectedness of Goetschel–Voxman fuzzy matroids (briefly, G–V fuzzy matroids), an analog of connectednessof crisp finite matroids. Based on the results of fuzzy circuits given by Goetschel and Voxman, the transitivity theorem concerningfuzzy circuits of G–V fuzzy matroids is established, and thus the useful notion of refined G–V fuzzy matroid is introduced. Theconnectedness of refined G–V fuzzy matroids is then defined by using an equivalence relation on the set of all fuzzy points on theground set, and some expected properties of these G–V fuzzy matroids are presented. Additionally, five kinds of fuzzy matroids arecompared.© 2010 Elsevier B.V. All rights reserved.

Keywords: G–V fuzzy matroid; Connectedness; Regular G–V fuzzy matroid; Refined G–V fuzzy matroid; Fuzzy circuit

1. Introduction and preliminaries

The notion of matroid was introduced by Whitney [24] in 1935 to provide a unifying abstract treatment of linearalgebra and graph theory. Actually, matroids play an important role in combinatorial optimization. A very interestingresult says that, for a given finite nonempty set X, a system of matroid independent sets on X is precisely a structurefor which the greedy algorithm works (see [20,23]). There have been two natural ways to extend the notion of matroidand related optimization problems to the fuzzy setting (see Section 5), one is using fuzzy intervals or something liketo instead of precise values in a matroid-based optimization problem (see [2,12,13]), the other is to fuzzify matroids.What we have involved in this paper is Goetschel–Voxman fuzzy matroids [3–10] (briefly, G–V fuzzy matroids), whichis in some degree a blueprint of all subsequent fuzzifications of matroids (see [11,21,22]).

Connectedness is a popular word to researchers, which is introduced in matroids as a generalization of 2-connectednessof graphs. This paper, as a continuation to the founding works [3–10], will study connectedness of G–V fuzzy matroids,actually an analog of connectedness of matroids. To do this, we first establish the transitivity theorem concerning fuzzycircuits of G–V fuzzy matroids in Section 2, based on which refined G–V fuzzy matroids can be defined in Section 3.

� This work was supported by the National Natural Science Foundation of China (Grant no. 10271069).∗Corresponding author.

E-mail addresses: [email protected], [email protected], [email protected], [email protected] (S.-g. Li).

0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2010.04.014


2710 X.-N. Li et al. / Fuzzy Sets and Systems 161 (2010) 2709–2723

In Section 4, we define the connected refined G–V fuzzy matroids, which are proved to have some useful properties ascrisp matroids. Apart from G–V fuzzy matroids, there are other definitions on fuzzy matroids. We compare five kindsof fuzzy matroids in Section 5 and conclude that G–V fuzzy matroids are worthy to be considered deeply because theypreserve many basic properties of crisp matroids. A concluding remark is given in the last section.

Now we present some fundamental notions and some useful results on matroids, G–V fuzzy matroids and fuzzy sets,which will be needed in this paper. Most of them are from [3–7,20,23] or are modifications to some notions or resultsin [3,5,23].

Definition 1.1 (Oxley [20], Welsh [23]). A matroid M is a pair (X, I) consisting of a finite set X and a nonempty familyI of subsets of X which satisfies the following two conditions:

(I1) If Y ∈ I and Z ⊆ Y , then Z ∈ I.(I2) If Y, Z ∈ I and 0 < |Y | < |Z | (where |Y | denotes the cardinality of Y), then there exists x ∈ Z − Y such that

Y ∪ {x} ∈ I.

One of the fundamental classes of matroids arise from graphs in the following way. For a graph G with an edge setX, let I = {Y ⊆ X |Y does not contain the edge set of a cycle of G}, then (X, I) is a matroid, called the cycle matroidof G and denoted by M(G).

Let M = (X, I) be a matroid. The members of I are called the independent sets of M. A set in I that is maximal in thesense of inclusion is called a base of the matroid M. A subset Y of X is called dependent if Y /∈ I. A minimal, in the senseof inclusion, dependent subset of X is called a circuit of the matroid M. We use the symbol B(M) (resp., C(M), D(M))to denote the family of all bases (resp., all circuits, all dependent sets) of the matroid M. Let B∗ = {X − B|B ∈ B(M)},then the dual matroid M∗ of M is the matroid on X whose bases are exactly the members of B∗. If {x} is a circuit, thenwe call {x} a loop. The rank function of the matroid M is a function R : 2X −→ N (the set of all natural numbers)defined by R(E) = max{|F ||F ⊆ E, F ∈ I}(∀E ∈ 2X ). A point x ∈ X is said to be dependent on a subset Y of X(denoted by x ∼ Y ) if and only if R(Y ) = R(Y ∪ {x}). For each Y ⊆ X , let Y− = {x ∈ X |x ∼ Y }, which is called theclosure of Y in (X, I).

The following basic property of circuits will be used in the subsequent discussion.

Proposition 1.2 (Oxley [20], Welsh [23]). Assume that M = (X, I) is a matroid, C1, C2 ∈ C(M), x1 ∈ C1 − C2,x2 ∈ C2 − C1 and C1 ∩ C2 �∅. Then there exists C3 ∈ C(M) such that x1, x2 ∈ C3 ⊆ C1 ∪ C2.

Definition 1.3 (Welsh [23]). A matroid M = (X, I) is said to be connected if for every pair of elements x and y of Xthere is a circuit of M which contains x and y.

Remark 1.4. Let M = (X, I) be a matroid. We define a relation ∼ between the elements of X by x ∼ y if and only ifx = y or there is a circuit C which contains both x and y. By Proposition 1.2, we know that∼ is an equivalence relationon X. It is easy to see that M is connected if and only if X is an equivalence class. Thus we get an alternative definitionof connected matroids (see, e.g. [20]).

Remark 1.5. Many graph-theoretic notions have matroid-theoretic analogues. Connectedness for matroids correspondsdirectly to the idea of 2-connectedness for graphs. The following result is well-known. Let G be a loopless graph withat least three vertices and without isolated vertices, then M(G) is connected if and only if G is 2-connected.

Let X be a finite set. A fuzzy set � on X is a mapping � : X −→ [0, 1]. We use [a] to denote the fuzzy set takingconstant value a on X (a ∈ [0, 1]) and �Y the characteristic function of Y (Y ⊆ X ). The fuzzy set [a] ∧ �{x}(a > 0)(usually denoted by xa) is called a fuzzy point. We denote the family of all fuzzy sets on X by [0, 1]X , and denote theset of all fuzzy points on X by J ([0, 1]X ). For any �, � ∈ [0, 1]X , we write �≤� if �(x)≤�(x) for each x ∈ X , and� < � if �≤� and � � �. � ∨ � and � ∧ � are fuzzy sets which satisfy (� ∨ �)(x) = max{�(x), �(x)} and (� ∧ �)(x) =min{�(x), �(x)}(∀x ∈ X ). For each � ∈ [0, 1]X , the nonnegative real number

∑x∈X �(x) (written as |�|) is called the

cardinality of �; we write supp� = {x ∈ X |�(x) > 0}, m(�) = min{�(x)|x ∈ supp �}, �[a] = {x ∈ X |�(x)≥a} (where0≤a≤1), and R+(�) = {�(x)|�(x) > 0, x ∈ X}.


X.-N. Li et al. / Fuzzy Sets and Systems 161 (2010) 2709–2723 2711

Definition 1.6 (Goetschel and Voxman [3,5]). Let X be a finite set, and � ⊆ [0, 1]X is a nonempty family of fuzzysets satisfying:

(FI1) If � ∈ �, � ∈ [0, 1]X and � < �, then � ∈ �.(FI2) If �, � ∈ � and 0 < |supp �| < |supp �|, then there exists � ∈ � such that

(i) � < �≤� ∨ �.(ii) m(�)≥min{m(�), m(�)}.

Then we call M = (X, �) a G–V fuzzy matroid, and call � the family of independent fuzzy sets of M. � ∈ � iscalled a fuzzy base if whenever � ∈ � and �≤� then � = �. � ∈ [0, 1]X is called a fuzzy circuit of M if � /∈ � and�\\x ∈ � for each x ∈ supp �, where �\\x is defined by

(�\\x)(y) ={

�(y), y � x,

0, y = x .

We use the symbols B(M) (resp., C(M)) to denote the family of all fuzzy bases (resp., fuzzy circuits) of M.

In the next proposition we see that each G–V fuzzy matroid induces a family of crisp matroids.

Proposition 1.7 (Goetschel and Voxman [3]). Let M = (X, �) be a G–V fuzzy matroid, and Ia = {�[a]|� ∈ �}(a ∈ (0, 1]). Then Ma = (X, Ia) is a crisp matroid on X, called the a-level matroid of (X, �).

Observation 1.8 (Goetschel and Voxman [3]). Suppose that M = (X, �) is a G–V fuzzy matroid, and that Ma =(X, Ia) is the crisp matroid on X defined in Proposition 1.7 (a ∈ (0, 1]). Then there is a finite sequence {a0, a1, . . . , an}such that

(i) 0 = a0 < a1 < · · · < an = 1 (such a sequence will be denoted by 〈a0, a1, . . . , an〉).(ii) If ai < a < b < ai+1(0≤i≤n − 1), then Ia = Ib.

(iii) If ai < a < ai+1 < b < ai+2(0≤i≤n − 2), then Ia ⊃ Ib.(iv) If 0 < a < b≤1, then Ia ⊇ Ib.

The above sequence a0, a1, . . . , an is called the fundamental sequence of M. A G–V fuzzy matroid M = (X, �)with a fundamental sequence 〈a0, a1, . . . , an〉 is said to be a closed G–V fuzzy matroid if whenever ai < a <

ai+1 (0≤i≤n − 1), then Ia = Iai+1 ; it is said to be regular if for any a, b ∈ [0, 1] with 0 < a < b≤1, and for eachbasis Y of (X, Ia), there is a base Z of (X, Ib) such that Z ⊆ Y .

The following proposition is a “partial converse” of Observation 1.8.

Proposition 1.9 (Goetschel and Voxman [3]). Let X be a finite set, and 0 = a0 < a1 < · · · < an = 1 a finite sequence.Suppose (X, Ia1 ), (X, Ia2 ), . . . , (X, Ian ) is a sequence of matroids on X satisfying Iai+1 ⊂ Iai (1≤i≤n − 1). For eacha ∈ (ai−1, ai ] (1≤i≤n), let Ia = Iai . Again, let � = {� ∈ [0, 1]X |�[a] ∈ Ia, 0 < a≤1}. Then M = (X, �) is a G–Vfuzzy matroid. Moreover, if M = (X, �) is a G–V fuzzy matroid, then � = {� ∈ [0, 1]X |�[a] ∈ Ia, 0 < a≤1}.

Remark 1.10. Every crisp matroid has at least one base, there are G–V fuzzy matroids which have no any fuzzy base.Closed G–V fuzzy matroids have fuzzy bases, while these fuzzy bases may have different cardinalities. This leads tothe notion of regularity. In [5], it was proved that all fuzzy bases of a closed regular G–V fuzzy matroid have the samecardinality. Since closed regular G–V fuzzy matroids have such an important property, many concepts, defined (suchas fuzzy dual, fuzzy matroid products, the greedy algorithm, etc.) or to be defined in this paper, are closely related toclosed regular G–V fuzzy matroids.

2. Transitivity of fuzzy circuits

Transitivity theorem on circuits is crucial for defining connectedness of crisp matroids. In this section, we will provethe transitivity theorem on fuzzy circuits of G–V fuzzy matroids (which is actually a fuzzy analog of Proposition 1.2).First we review some preliminaries.



Definition 2.1 (Goetschel and Voxman [5]). The truncation of a fuzzy set � at a level a is the fuzzy set �Ta , defined by

�Ta (x) =

{�(x) if �(x)≤a,

a if �(x) > a.

If M = (X, �) is a G–V fuzzy matroid and � ∈ [0, 1]X , then we define � : [0, 1]X −→ [0, 1] by �(�) = max{a ∈[0, 1]|�T

a is not a fuzzy circuit in M}. The mapping � is called the circuit function for �.

Remark 2.2. In the original text it reads 0 when �(x) > a instead of a, which is a typing error, and it misses the word“fuzzy” in the expression of �(�).

Definition 2.3 (Goetschel and Voxman [5]). Suppose that M is a G–V fuzzy matroid and that � is a fuzzy circuit inM. Let C(�) = (�(�), m(�)] if �(�) < m(�) and C(�) = {m(�)} if �(�) = m(�), then we call C(�) the circuit intervalof �.

Lemma 2.4 (Goetschel and Voxman [5]). Suppose that M = (X, �) is a G–V fuzzy matroid, and that � is a fuzzycircuit in M.

(i) If a ∈ C(�), then �[a] is a circuit in the crisp matroid Ma = (X, Ia).(ii) If a /∈ C(�), then �[a] ∈ Ia .

Remark 2.5. Let � be a fuzzy circuit in a G–V fuzzy matroidM, then it is easy to prove �(�)≤m(�). IfM is a closedG–V fuzzy matroid, then �(�) < m(�). In [5], circuit intervals were defined only for closed G–V fuzzy matroids.Definition 2.3 has been modified to drop this restriction. It is easy to check that Lemma 2.4 also holds under thismodification.

Lemma 2.6 (Goetschel and Voxman [5]). Suppose that M = (X, �) is a G–V fuzzy matroid and that � ∈ [0, 1]X .Let R+(�) = 〈b1, b2, . . . , bk〉. Then � is a fuzzy circuit in M if and only if it satisfies the following conditions:

(i) �[b1] is a circuit in the crisp matroid (X, Ib1 ).(ii) �[b j ] ∈ Ib j , 2≤ j≤k.

Theorem 2.7 (Transitivity theorem). Let (X, �) be a G–V fuzzy matroid, xd1 , yd2 , zd3 ∈ [0, 1]X , and �, � ∈ C(M).If xd1 ∨ yd2≤�, yd2 ∨ zd3≤� and C(�) ∩ C(�) �∅, then there is a fuzzy circuit � such that xd1 ∨ zd3≤�.

Proof. Suppose that R+(�) = 〈b1, b2, . . . , bm〉, R+(�) = 〈c1, c2, . . . , cn〉. Without loss of generality, we assumeb1≤c1. We will construct a fuzzy circuit � which satisfies xd1 ∨ zd3≤�. Since C(�) ∩ C(�) �∅, it follows from thedefinition of the circuit interval that b1 = m(�) ∈ C(�). By Lemma 2.4, �[b1] and �[b1] are circuits in the crisp matroid(X, Ib1 ). Observe that xd1≤�, yd2≤�, so x, y ∈ �[b1]. Similarly, y, z ∈ �[c1] = �[b1]. By Proposition 1.2, there exists acircuit C in (X, Ib1 ) such that x, z ∈ C . Let �′ = (�C ∧ [b1])∨ xd1 ∨ zd3 . We will construct the required fuzzy circuit� by considering the following four cases.

Case 1: b1 < d1 < d3 or b1 < d3 < d1. Without loss of generality, we assume b1 < d1 < d3. Then �′[b1] = C isa circuit in the crisp matroid (X, Ib1 ). First, as d1 > b1 = m(�), �[d1] ∈ Id1 by Lemma 2.4; moreover {x} ⊆ �[d1],therefore {x} ∈ Id1 . Next, we show {z} ∈ Id1 . If d3 /∈ C(�), then {z} ⊆ �[d3] ∈ Id3 , thus {z} ∈ Id3 . If d3 ∈ C(�),observe that d3≤c1, hence {y, z} ⊆ �[d3]. As �[d3] is a circuit in (X, Id3 ), {z} ∈ Id3 . Since d1 < d3, Id3 ⊆ Id1 by (iv) ofObservation 1.8, and thus {z} ∈ Id1 . It follows from the above proved facts {x} ∈ Id1 and {z} ∈ Id1 that {x, z} ∈ Id1

or {x, z} is a circuit in (X, Id1 ). If {x, z} ∈ Id1 , then �′[d1] = {x, z} ∈ Id1 and �′[d3] = {z} ∈ Id3 . By Lemma 2.6, �′

is a fuzzy circuit. Obviously zd3≤�′ and xd1≤�′, and thus we take � = �′. If {x, z} is a circuit in the crisp matroid(X, Id1 ), we take � = xd1 ∨ yd3 . By Lemma 2.6, � is a fuzzy circuit, and zd3≤�, xd1≤�.

Case 2: d1 = d3 > b1. The proof is similar to Case 1.Case 3: d3≤b1 < d1 or d1≤b1 < d3. Without loss of generality, we suppose that d3≤b1 < d1. Then �′ =

(�C ∧ [b1]) ∨ xd1 . We take � = �′. It is easy to check that � is a fuzzy circuit such that xd1≤� and zd3≤�.Case 4: d1≤b1 and d3≤b1. Then �′ = [b1] ∧ �C . We take � = �′. By Lemma 2.6, � ∈ C(M). Obviously, zd3≤�

and xd1≤�, as desired. �



The following example shows that the condition C(�) ∩ C(�) �∅ in Theorem 2.7 cannot be deleted.

Example 2.8. Let X = {x, y, z}, I1/2 = {∅, {x}, {y}, {z}, {x, y}, {x, z}}, and I1 = {∅, {x}, {z}}. Then (X, I1/2) and(X, I1) are crisp matroids and I1 ⊂ I1/2. Let

Ia ={ I1/2, 0 < a≤ 1

2 ,

I1,12 < a≤1,

� = {� ∈ [0, 1]X |�[a] ∈ Ia, 0 < a≤1}, � = x1∨ z1, and � = y1/2∨ z1. ThenM = (X, �) is a G–V fuzzy matroid byProposition 1.9, and �, � ∈ C(M) and C(�) ∩ C(�) = ∅, while there is no fuzzy circuit � which satisfies x1, y1/2≤�.

Remark 2.9. Suppose thatM = (X, �) is a G–V fuzzy matroid and �(�) = 0 for each � ∈ C(M). Then C(�)∩C(�) �∅for any �, � ∈ C(M), and transitivity theorem holds forM. We can imitate the method in Remark 1.4 to define connectedG–V fuzzy matroids on this special class of G–V fuzzy matroids. For every G–V fuzzy matroid in this class, we canprove |{Ia |a ∈ (0, 1]}|≤2. In [14], (X, �) is said to be singular 1 (resp., single) if |{Ia |a ∈ (0, 1]}| = 1 (resp., 2).Li et al. [14] concerned mainly about closed singular G–V fuzzy matroids, and they defined a closure operator whichcan determine closed singular G–V fuzzy matroids. While singular G–V fuzzy matroids are particularly simple. Weshall define refined G–V fuzzy matroids in the next section and focus our attention on the larger class of G–V fuzzymatroids.

3. Refined G–V fuzzy matroids

Based on Section 2, we introduce a kind of special G–V fuzzy matroids, called refined G–V fuzzy matroids, in thissection, which is suitable for defining our connectedness on G–V fuzzy matroids.

Definition 3.1 (Bean [1]). A matroid M2 is said to be a refinement of a matroid M1 if they are defined on a commonset X and each circuit of M1 is a circuit of M2.

Definition 3.2. A G–V fuzzy matroid M = (X, �) is said to be refined if (X, Ib) is a refinement of (X, Ia) for anya, b ∈ (0, 1] satisfying a < b.

Example 3.3. Let G1, G2 and G3 be the graphs shown in Fig. 1. Then the corresponding cycle matroids are M1 =(X, I1), M2 = (X, I2) and M3 = (X, I3), respectively, where X = {e1, e2, e3, e4}, I1 = 2X , I2 = 2{e1,e3,e4} ∪2{e2,e3,e4}, and I3 = I2 − {{e1, e3, e4}, {e2, e3, e4}}. Let

Ia =

⎧⎪⎨⎪⎩I1, 0 < a≤ 1

3 ,

I2,13 < a≤ 2

3 ,

I3,23 < a≤1,

and � = {� ∈ [0, 1]X |�[a] ∈ Ia, 0 < a≤1}, thenM = (X, �) is a G–V fuzzy matroid by Proposition 1.9. Obviously,C(M1) = ∅, C(M2) = {{e1, e2}}, and C(M3) = {{e1, e2}, {e1, e3, e4}, {e2, e3, e4}}. Thus M is a refined G–V fuzzymatroid.

Now we shall prove an important property of refined G–V fuzzy matroids.

Definition 3.4 (Novak [17]). Let M1 and M2 be a pair of distinct matroids on the same ground set X. We say that M2is regular in M1 if for each base B1 ∈ B(M1), there is a base B2 ∈ B(M2) such that B2 ⊆ B1.

Proposition 3.5 (Oxley [20], Welsh [23]). Let M be a matroid on X. If B is a base of M and x ∈ X − B, then thereexists a unique circuit CM (x, B) such that x ∈ CM (x, B) ⊆ B ∪ {x}. This circuit CM (x, B) is called the fundamentalcircuit of x with respect to B.

1 In [8], (X, �) is called an elementary G–V fuzzy matroid if |{Ia |a ∈ (0, 1]}| = 1.



e1

e2

e3

e4

e1

e2

e3

e4

e1

e2

e3e4

G1 G2 G3

Fig. 1.

Proposition 3.6. Let M1 and M2 be a pair of distinct matroids defined on a common set X. If M2 is a refinement ofM1, then M2 is regular in M1.

Proof. Since M2 is a refinement of M1 and they are distinct matroids defined on X, then C(M1) ⊂ C(M2). Clearly,{Y ⊆ X | there exists Z ∈ C(M1) such that Z ⊆ Y } ⊂ {Y ⊆ X | there exists Z ∈ C(M2) such that Z ⊆ Y }, that isD(M1) ⊂ D(M2). Therefore, 2X − D(M2) ⊂ 2X − D(M1), that is I(M2) ⊂ I(M1). Assume that A ∈ B(M1), weneed to show B ⊆ A for some B ∈ B(M2). If A ∈ I(M2), then A ∈ B(M2) for I(M2) ⊂ I(M1). Thus we supposeA ∈ D(M2). Let B be a maximal independent set in M2 such that B ⊆ A. It suffices to show B ∈ B(M2). Assumethat B /∈ B(M2), i.e. B ∪ C ∈ B(M2) for some nonempty set C ⊆ X − A. Since I(M2) ⊂ I(M1), B ∪ C ∈ I(M1).From (I2) of Definition 1.1, there exists a D∗ ⊆ A − B such that C ∪ B ∪ D∗ ∈ B(M1). Let C ∪ D∗ = D. Since|A| = |B ∪ D|, |A − B ∪ D| = |C |. As C �∅, we can choose x ∈ A − B ∪ D. It follows from Proposition 3.5 thatthere exists a circuit CM1 (x, B ∪ D) such that x ∈ CM1 (x, B ∪ D) ⊆ {x} ∪ (B ∪ D). Clearly, C ∩CM1 (x, B ∪ D) �∅.We consider the following two cases:

Case 1: D∗ ∩CM1 (x, B ∪ D) = ∅. Observe that x ∈ CM1 (x, B ∪ D) ⊆ {x} ∪ B ∪C , CM1 (x, B ∪ D) ∈ C(M2), andB ∪C ∈ B(M2). By the uniqueness of CM2 (x, B ∪C), CM1 (x, B ∪ D) = CM2 (x, B ∪C). Let y ∈ C ∩CM2 (x, B ∪C),since CM2 (x, B ∪ C) is the unique circuit contained in {x} ∪ (B ∪ C), (B ∪ C − {y}) ∪ {x} ∈ I(M2). Notice that|(B ∪ C − {y}) ∪ {x}| = |B ∪ C |, thus (B ∪ C − {y}) ∪ {x} ∈ B(M2), and x ∪ B ∈ I(M2), which contradicts the factthat B is the maximal independent set contained in A.

Case 2: D∗ ∩ CM1 (x, B ∪ D) = {y1, y2, . . . , yn} is an n-element set (n≥1). As y1 /∈ B ∪ C, B ∪ C ∈ B(M2),there exists CM2 (y1, B ∪ C) ∈ C(M2) such that y1 ∈ CM2 (y1, B ∪ C) ⊆ {y1} ∪ B ∪ C . Since x ∈ CM1 (x, B ∪D) − CM2 (y1, B ∪ C), and y1 ∈ CM1 (x, B ∪ D) ∩ CM2 (y1, B ∪ C), there exists a circuit C1 ∈ C(M2) such thatx ∈ C1 ⊆ CM1 (x, B ∪ D) ∪ CM2 (y1, B ∪ C) − {y1} by the strong circuit elimination axiom.2 After finite steps, wecan get a circuit Cm ∈ C(M2) which satisfies x ∈ Cm ⊆ {x} ∪ (B ∪ D) and Cm ∩ D∗ = ∅. Using Cm to replaceCM1 (x, B ∪ D) in Case 1, we still have a contradiction. �

Proposition 3.6 implies

Proposition 3.7. Every refined G–V fuzzy matroid is regular.

Remark 3.8. (1) The converse of Proposition 3.7 is not true. In fact, the G–V fuzzy matroid M in Example 2.8 isregular, but it is not a refined G–V fuzzy matroid.

(2) Let M be a closed regular G–V fuzzy matroid with a fundamental sequence 〈a0, a1, . . . , an〉, B the family offuzzy bases of M, and B∗ = {�c|� ∈ B} (where �c = [1]− �). Then B∗ forms the family of fuzzy bases for a closedregular G–V fuzzy matroidM∗.M∗ is called the dual ofM (see [7]), and its fundamental sequence is 〈t0, t1, . . . , tn〉,where ti = 1− an−i (i = 1, 2, . . . , n). If M is a refined G–V fuzzy matroid, then M∗ is well-defined by Proposition3.7. However, M∗ needs not to be a refined G–V fuzzy matroid (see Example 3.9).

2 I.e., if C1 and C2 are circuits such that x ∈ C1 ∩ C2 and y ∈ C1 − C2, then there exists a circuit C3 such that y ∈ C3 ⊆ C1 ∪ C2 − {x}. See,for example, [20,23].



Example 3.9. Let X = {1, 2, 3, 4},B1/2 = {{1, 3}, {1, 4}}, andB1 = {{1}, {3}, {4}}. Again, let (X, I1/2) and (X, I1) becrisp matroids having the families of bases B1/2 and B1 respectively.3 Then we can check that C(M1/2) = {{3, 4}, {2}}and C(M1) = {{1, 3}, {1, 4}, {3, 4}, {2}}. Let

Ia ={ I1/2, 0 < a≤ 1

2 ,

I1,12 < a≤1,

and � = {� ∈ [0, 1]X |�[a] ∈ Ia, 0 < a≤1}, then M = (X, �) is a refined G–V fuzzy matroid. The fundamentalsequence ofM∗ is 〈0, 1

2 , 1〉, D1/2 = M∗1 (where D1/2 is the 12 -level matroid ofM∗, cf. [7]) and D1 = M∗1/2. Observe

that {1, 3, 4} ∈ C(D1/2), but {1, 3, 4} /∈ C(D1). Therefore, M∗ is not a refined G–V fuzzy matroid.

The next result shows an important property of refined G–V fuzzy matroids.

Proposition 3.10. Let M = (X, �) be a refined G–V fuzzy matroid, xa, yb, zc ∈ [0, 1]X , and �, � ∈ C(M) satisfyingxa ∨ yb≤� and yb ∨ zc≤�. Then there exist �1, �1 ∈ C(M) such that xa ∨ yb≤�1, yb ∨ zc≤�1, and C(�1) ∩ C(�1) �∅.

Proof. Without loss of generality, we suppose m(�)≤m(�). Let �1 = [m(�)] ∧ ��[m(�)], and

�2 =∨

d∈R+(�)−{m(�)}([d] ∧ ��[d]

).

Then we can check that �1 ∈ C(M), �2 ∈ �, supp�2 ⊆ supp �1, and � = �1 ∨ �2. Similarly, let � = �1 ∨ �2 (where�1 ∈ C(M), �2 ∈ �), then supp�2 ⊆ supp �1. We consider the following two cases:

Case 1: b≤m(�). Let �1 = ([m(�)] ∧ ��[m(�)]) ∨ �2, and �1 = �.

Case 2: b > m(�). Let �1 = ([b] ∧ ��[m(�)]) ∨ �2, and �1 = ([b] ∧ ��[m(�)]

) ∨ �2.

Note that (X, �) is a refined G–V fuzzy matroid, we can check that �1, �1 are fuzzy circuits, C(�1) ∩ C(�1) �∅ andxa ∨ yb≤�1, yb ∨ zc≤�1, as desired. �

From Theorem 2.7 and Proposition 3.10, we have

Theorem 3.11. Let M = (X, �) be a refined G–V fuzzy matroid and xa, yb, zc ∈ [0, 1]X . Suppose that �, � ∈ C(M)satisfying xa ∨ yb≤� and yb ∨ zc≤�, then xa ∨ zc≤� for some � ∈ C(M).

4. Connectedness of refined G–V fuzzy matroids

In this section, we will define a kind of connectedness in refined G–V fuzzy matroids and present some usefulproperties of them.

Let M = (X, �) be a refined G–V fuzzy matroid. For every pair of fuzzy points xa and yb, define xa ∼ yb iffxa = yb or xa, yb≤� for some � ∈ C(M). By Theorem 3.11,∼ is an equivalence relation on J ([0, 1]X ). If there is onlyone equivalence class determined by this equivalence relation, we call M a connected G–V fuzzy matroid. Supposethat � ∈ [0, 1]X . If for every pair of fuzzy points xa, yb≤�, there exists � ∈ C(M) such that xa, yb≤�≤�, then � is saidto be connected in M.

Remark 4.1. (1) LetM = (X, �) be a G–V fuzzy matroid. It seems feasible to imitate Definition 1.3 to defineM tobe connected if M satisfies the following condition:

(C) For every pair of fuzzy points xa and yb, there exists � ∈ C(M) such that xa ∨ yb≤�.

We do not do this mainly because M satisfying (C) need not satisfy the following condition (see Example 4.2):

(E) There exists an a-level matroid Ma of M such that Ma is connected.

3 It is easy to check that B1/2 and B1 satisfy the base axioms: a nonempty family B of subsets of X is the set of bases of a matroid on X if and onlyif it satisfies the condition: (B1) If B1, B2 ∈ B and x ∈ B1 − B2, then there exists y ∈ B2 − B1 such that B1 ∪ {y} − {x} ∈ B.



(2) Remark 1.4 introduces an alternative definition of connected matroids. This definition is based on Proposition 1.2.To use this method to define connected G–V fuzzy matroids, we need to find a certain class of G–V fuzzy matroids forwhich the fuzzy analog of Proposition 1.2 holds. Refined G–V fuzzy matroids are such a class of G–V fuzzy matroidsand can be easily constructed (see Example 3.3).

We restate the two conditions imposed on M as follows:

(R) If 0 < a < b≤1, then (E, Ib) is a refinement of (E, Ia).(T) If �, � ∈ C(M) satisfying xa ∨ yb≤� and yb ∨ zc≤�, then xa ∨ yb≤� for some � ∈ C(M).

The relations between statements (C), (E), (R) and (T) are illustrated as follows (where “−→” and “��” mean“implies” and “does not imply” respectively):

(T) (C)

(E)

(R)

(C)→ (T) is obvious, and (R)→ (T) comes from Theorem 3.11. We give the following counterexample to show(C) �� (E); counterexamples for other cases are easy to give.

Example 4.2. Let X = {1, 2, 3, 4, 5}, B1/3 = {{1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 4, 5}}, B2/3 = {{1, 2, 5},{1, 3, 5}, {1, 4, 5}, {2, 4, 5}, {3, 4, 5}}, and B1 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}}. Suppose that (X, I1/3), (X, I2/3) and(X, I1) are crisp matroids on X having the families of bases B1/3, B2/3 and B1, respectively.4 Let

Ia =

⎧⎪⎨⎪⎩I1/3, 0 < a≤ 1

3 ,

I2/3,13 < a≤ 2

3 ,

I1,23 < a≤1,

and � = {� ∈ [0, 1]X |�[a] ∈ Ia, 0 < a≤1}. Then M = (X, �) is a G–V fuzzy matroid. We can check thatM = (X, �) satisfies (C), but none of the three level crisp matroids is connected.

The following result characterizes a connected G–V fuzzy matroid in terms of connected crisp matroids.

Proposition 4.3. Let M = (X, �) be a refined G–V fuzzy matroid, then M is connected if and only if (X, I1) is aconnected matroid.

Proof. Necessity. Let x, y ∈ X with x � y. Since M is connected, there exists a � ∈ C(M) satisfying x1 ∨ y1≤�. ByLemma 2.4, �[m(�)] is a circuit in the crisp matroid (X, Im(�)). By definition of refined G–V fuzzy matroids, �[m(�)] isa circuit in (X, I1). Therefore (E, I1) is connected.

Sufficiency. Let xa, yb ∈ [0, 1]X . Since (X, I1) is connected, there exists a circuit C in (X, I1) satisfying x, y ∈ C .Let � = �C , then � ∈ C(M) and xa, yb≤�, so M is connected. �

Example 4.4. Let M be the G–V fuzzy matroid in Example 3.3. Note that G3 in Fig. 1 is 2-connected, so thecorresponding cycle matroid M3 is connected. By Proposition 4.3, M is connected.

The following results involving connectedness are well-known in crisp matroids (see, e.g., [20,23]), we will examinefuzzy analogs of these results.

4 See footnote 3.



Proposition 4.5. Suppose that M = (X, I) is a crisp matroid.

(i) M is connected if and only if M∗ is connected.(ii) If Y, Z are connected subsets of M and Y ∩ Z �∅, then Y ∪ Z is connected.

(iii) Assume that M has no loop and Y ⊆ X . If there exists a connected subset Z such that Z ⊆ Y ⊆ Z−, then Y(particularly, Z−) is connected.

(iv) M is not connected if and only if there exists a proper nonempty subset Y of M such that R(Y )+ R(X−Y ) = R(X ).

Suppose thatM is a refined G–V fuzzy matroid, we have already known thatM∗ need not be a refined G–V fuzzymatroid. If both M and M∗ are refined G–V fuzzy matroids, then we have the following result.

Proposition 4.6. Assume that both M and M∗ are refined G–V fuzzy matroids, then M is connected if and only ifM∗ is connected.

Proof. SinceM∗∗ =M, it suffices to show the necessity. Let xa, yb ∈ [0, 1]X , then there exists � ∈ C(M) satisfyingxa∨ yb≤�, thus x, y ∈ �[m(�)] and �[m(�)] is a circuit in (X, I1). By Proposition 4.2.9 in [20], there exists C∗ ∈ C(M∗1 )such that C∗ ∩ �[m(�)] = {x, y}. Obviously, �C∗ ∈ C(M∗) and xa ∨ yb≤�C∗ , this concludes the proof. �

The following example shows that the fuzzy analog of part (ii) of Proposition 4.5 does not hold.

Example 4.7. LetM be the refined G–V fuzzy matroid defined in Example 3.3, � = �{e2,e3,e4}, and � = �{e1,e2} ∧ [ 12 ].

We can check that � and � are connected and � ∧ � � [0], but � ∨ � is not connected.

Under certain conditions, we can establish a fuzzy analog of part (ii) of Proposition 4.5.

Proposition 4.8. Suppose that M = (X, �) is a refined G–V fuzzy matroid with a fundamental sequence〈a0, a1, . . . , an〉, and �, � are connected fuzzy sets. If there exists z ∈ X such that m(�) = �(z), m(�) = �(z) and�(z), �(z) ∈ (a j , a j+1) for some j ∈ {0, 1, . . . , n − 1}, then � ∨ � is connected.

Proof. Let xa , yb≤� ∨ �. It is clear that we need only to consider the case xa≤� and yb � � and the case yb≤� andyb � �. Without loss of generality, we assume that �(z)≤�(z). Since �, � are connected, there exist �1, �2 ∈ C(M)satisfying xa ∨ z�(z)≤�1≤� and yb ∨ z�(z)≤�2≤�. Obviously, C(�1)∩ C(�2) �∅. It follows from Theorem 2.7 and itsproof that there exists a fuzzy circuit �3 such that xa ∨ yb≤�3≤�1 ∨ �2≤� ∨ �. �

In order to establish a fuzzy analog of part (iii) of Proposition 4.5, we need the following definition and result.

Definition 4.9 (Goetschel and Voxman [8], Li et al. [14]). Let M = (X, �) be a G–V fuzzy matroid. The function : [0, 1]X → [0,∞) defined by (�) = sup{|�||�≤�, � ∈ �} is called the rank function of M. The fuzzy set�− =∨{xa ∈ [0, 1]X |(�) = (� ∨ {xa})} is called the closure of �.

Proposition 4.10 (Li et al. [14]). Suppose that M = (X, �) is a G–V fuzzy matroid. Let x ∈ X and � ∈ [0, 1]X . Fora given a ∈ (0, 1], if xa≤�− and xa � �, then x ∈ Ca ⊆ �[a] ∪ {x} for some circuit Ca in (X, Ia).

Proposition 4.11. Let M = (X, �) be a refined G–V fuzzy matroid with a fundamental sequence 〈a0, a1, . . . , an〉,and (X, I1) has no loop. Assume that � is a connected fuzzy set and m(�)≤a1, then every fuzzy set � with �≤�≤�−(particularly, �−) is connected.

Proof. Let xa, yb≤�. We will construct a fuzzy circuit � which satisfies xa ∨ yb≤�≤�.Case 1: �(x), �(y) > 0. Let c = min{�(x), �(y), a, b}, then xc∨ yc≤�. Since � is connected, there exists �∗ ∈ C(M)

satisfying xc ∨ yc≤�∗≤�. Therefore, x, y ∈ �∗[m(�∗)]. By Lemma 2.4, �∗[m(�∗)] is a circuit of (X, Im(�∗)). Without lossof generality, we suppose a≤b. Since M is a refined G–V fuzzy matroid and (X, I1) has no loop, {x, y} is either acircuit or an independent set of (X, Ia). If {x, y} is a circuit, then we take � = xa ∨ yb. If {x, y} is an independent set,



then we take

� = ([m(�∗)] ∧ ��∗[m(�∗)]) ∨ xa ∨ yb.

Note that (X, I1) has no loop, it follows from definition of the refined G–V fuzzy matroid that {y} is an independentset of (X, Ib). Obviously, � ∈ C(M) and xa ∨ yb≤�≤�.

Case 2: �(x) = 0 or �(y) = 0. Without loss of generality, we assume that �(x) = 0. Let c = min{a1, a}, we shallprove that � ∨ xc is connected. Let xc, we≤� ∨ xc. By Proposition 4.10, there exists a circuit C ∈ C(Mc) satisfyingx ∈ C ⊆ (� ∨ xc)[c]. As (X, I1) has no loop, so does (X, Ia1 ). Then C ∩ �[c] �∅. Let z ∈ C ∩ �[c]. If w ∈ C , thenwe take � = ([c] ∧ �C ) ∨ we. If w /∈ C , consider wc and um(�) (where m(�) = �(u)). Since wc ∨ um(�)≤� and � isconnected, there exists a fuzzy circuit v which satisfies wc ∨ um(�)≤v≤�. Therefore, we can choose a circuit C1 of(X, Ia1 ) satisfying w, u ∈ C1. Similarly, we can choose a circuit C2 of (X, Ia1 ) which satisfies z, u ∈ C2. Note thatC is a circuit of (X, Ia1 ) and x, z ∈ C , by Proposition 1.2, it is easy to see that there exists a circuit C3 of (X, Ia1 )which contains both x and w. Let n = min{e, c} and � = ([n] ∧ �C3

) ∨ (we ∨ xc). Then � ∈ C(M), xc ∨ we≤� and�≤� ∨ xc. So � ∨ xc is connected. If �(y) = 0, similarly, � ∨ xc ∨ yd is connected (where d = min{c, b}). It followsfrom Case 1 that � is connected. �

We conclude this section by giving a fuzzy analog of part (iv) of Proposition 4.5.

Proposition 4.12 (Goetschel and Voxman [3]). Suppose that M = (X, �) is a G–V fuzzy matroid with a funda-mental sequence 〈a0, a1, . . . , an〉. For each � ∈ [0, 1]X with R+(�) = 〈b1, b2, . . . , bm〉, let 〈c0, c1, . . . , c j 〉 ={a1, a2, . . . , am} ∪ {b0, b1, . . . , bn}. Then

(�) =k= j∑k=1

(ck − ck−1)Rck (�[ck ]),

where Rck is the rank function of (E, Ick ) (k = 1, 2, . . . , j).

Proposition 4.13. Assume that M = (X, �) is a refined G–V fuzzy matroid with a fundamental sequence〈a0, a1, . . . , an〉. Then M is not connected if and only if there exists a proper nonempty subset Y of X such that(�Y )+ (�X−Y ) = (M).

Proof. Necessity. Suppose thatM is not connected. By Proposition 4.3, (X, I1) is not connected. SinceM is a refinedG–V fuzzy matroid, (X, Iai ) is not connected (i = 1, 2, . . . , n). It follows from (iv) of Proposition 4.5 that there exists aproper nonempty subset Yi of X such that Rai (Yi )+ Rai (X−Yi ) = Rai (X ) (i = 1, 2, . . . , n). This means that, for everycircuit C ∈ C(Mai ), C ⊆ Yi or C ⊆ X − Yi . Since M is a refined G–V fuzzy matroid, for every circuit C ∈ C(Mai ),C ⊆ Y1 or C ⊆ X − Y1. Thus Rai (Y1)+ Rai (X − Y1) = Rai (X )(i = 1, 2, . . . , n). By Proposition 4.12, we have

(M)=k=n∑k=1

(ak − ak−1)Rrk (X ) =k=n∑k=1

(ak − ak−1)(Rak (Y1)+ Rak (X − Y1))

=k=n∑k=1

(ak − ak−1)Rak ((�Y1)[ak ])+

k=n∑k=1

(ak − ak−1)Rak ((�X−Y1)[ak ]) = (�Y1

)+ (�X−Y1).

Sufficiency. Since (�Y )+ (�X−Y ) = (M), we have

k=n∑k=1

(ak − ak−1)Rak ((�Y1)[ak ])+

k=n∑k=1

(ak − ak−1)Rak ((�X−Y1)[ak ]) =

k=n∑k=1

(ak − ak−1)Rak ((�X )[ak ])

and thus

k=n∑k=1

(ak − ak−1)(Rak (X − Y1)+ Rak (Y1)− Rak (X )) = 0.



As ak − ak−1 > 0 and Rak (X − Y1)+ Rak (Y1)− Rak (X )≥0, Rak (X − Y1)+ Rak (Y1)− Rak (X ) = 0 (k = 1, 2, . . . n).By (iv) of Proposition 4.5, (X, I1) is not connected. It follows from Proposition 4.3 that M is not connected, thisconcludes the proof. �

5. Comparison of five kinds of fuzzy matroids

As in the case of fuzzy topology, there are other definitions on fuzzy matroids (apart from G–V fuzzy matroids). In thissection, we will give a comparison of five kinds of fuzzy matroids, which may be helpful for studying connectednessof fuzzy matroids and related greedy algorithms. First, let us recall other two kinds of fuzzy matroids which wereintroduced in [3].

Definition 5.1 (Goetschel and Voxman [3, Definition 4.1]). Suppose that X is a finite set and that � ⊆ [0, 1]X is anonempty family of fuzzy sets satisfying:

(FI1) If � ∈ �, � ∈ [0, 1]X and � < �, then � ∈ �.(FI2) If �, � ∈ �, 0 < |supp �| < |supp �|, and m(�)≤m(�), then there exists � ∈ � such that

(i) � < �≤� ∨ �.(ii) m(�)≥m(�).

Then we call M = (X, �) a weak fuzzy matroid.

Definition 5.2 (Goetschel and Voxman [3, Definition 4.3]). Suppose that X is a finite set and that � ⊆ [0, 1]X is anonempty family of fuzzy sets satisfying:

(FI1) If � ∈ �, � ∈ [0, 1]X and � < �, then � ∈ �.(FI2) If �, � ∈ � and |�| < |�|, then there exists � ∈ � such that

(i) �≥�.(ii) |�| < |�|≤|�|.

(iii) � \ �≤� \ �, where (� \ �)(x) = max{�(x)− �(x), 0} (∀x ∈ X ).

Then we call M = (X, �) a fuzzy matroid.

Remark 5.3. (1) Every G–V fuzzy matroid is a weak fuzzy matroid, but not vice versa (see Example 5.9).(2) There is no implication between the notions G–V fuzzy matroid and fuzzy matroid (see Examples 5.9 and 5.11).(3) As pointed out in [3], the principal defect of weak fuzzy matroids is that their rank functions are not necessarily

submodular. The problem with Definition 5.2 is that many of the basic properties of matroids (including submodularity)do not carry over to the fuzzy case. These are barriers for studying fuzzy connectedness which is an analog or fuzzificationof connectedness of crisp matroids.

These three kinds of fuzzy matroids mentioned above have one common denominator, namely, they are all based onthe straight extension of the independence axioms from set systems to fuzzy set systems. Note if a fuzzy matroid onX is to be defined as a pair (X, �), where � is a nonempty family of fuzzy subsets of X satisfying some conditions,then this fuzzy matroid should be equivalent to some structure on the set [0, 1]X . Since [0, 1]X is an infinite set, fuzzymatroids on X should be members of the class of infinite matroids (see, e.g., [23, Chapter 20]) on [0, 1]X . In [11], onepossible fuzzification of matroids based on this idea is proposed as follows:

Definition 5.4 (Hsueh [11, Definition 3.2]). Suppose that X is a finite set and that � ⊆ [0, 1]X is a nonempty familyof fuzzy sets satisfying:

(FI1) If � ∈ �, � ∈ [0, 1]X and � < �, then � ∈ �.(FI2) If �, � ∈ � and |�| < |�|, then there exists an element x ∈ X and a fuzzy subset � ∈ � such that �(y) = �(y)

(∀y ∈ X − {x}) and �(x) < �(x)≤�(x).(FI3) Let � be a fuzzy subset of X. If {� ∈ [0, 1]X |� < �} ⊆ �, then � ∈ �.



Then we callM = (X, �) an H fuzzy matroid, call the numbers of � independent fuzzy sets ofM, and call a maximalindependent fuzzy set a fuzzy base of M.

H fuzzy matroids and G–V fuzzy matroids are closely related. First, we have the following

Proposition 5.5. For a G–V fuzzy matroid (X, �), the following statements are equivalent:

(1) (X, �) is an H fuzzy matroids.(2) (X, �) has one fuzzy base.

Proof. (1) �⇒ (2) Let 〈a0, a1, . . . , an〉 be the fundamental sequence of (X, �). By Proposition 1.9, it is easy to provethat for every fuzzy base � of the G–V fuzzy matroid (X, �), R+(�) ⊆ {a0, a1, . . . , an} (cf. [4, Theorem 1.9]). Thus(X, �) has finitely many fuzzy bases. Let X = {x1, x2, . . . , xl} and we assume that (X, �) has n (n≥2) fuzzy bases,and derive a contradiction. Let �M be the family of fuzzy bases of H fuzzy matroid (X, �), and let x M

i = {�(xi ) >

0|� ∈ �M } (i = 1, 2, . . . , l). Since (X, �) has n (n≥2) fuzzy bases, there exist �1, �2 ∈ �M and j ∈ {1, 2, . . . , l} suchthat �1(x j ) = max x M

j > 0, �2(x j ) = max(x Mj −{�1(x j )}) (note that �2(x j ) may be 0). Let �∗1 ∈ [0, 1]X be defined by

�∗1(x) ={

(�1(e j )+ �2(x j ))/2, x j ,

�1(x) otherwise.

Note that |�∗1| < |�1| = |�2|, it follows from (FI2) of Definition 5.4 that there exist y ∈ X and � ∈ � satisfying�(x) = �∗1(x) for all x ∈ X−{y} and �∗1(y) < �(y)≤�2(y). Let �3 ∈ �M and �≤�3. Obviously, for all x ∈ X−{y, x j },�3(x)≥�(x) = �1(x), �3(x j ) = �1(x j ) and �3(y)≥�(y) > �∗1(y) = �1(y), thus we have |�1| < |�3|. It contradictsthe fact that all fuzzy bases of an H fuzzy matroid have the same cardinality (see [11, Theorem 2.1]). Obviously, (2)implies (1). �

Remark 5.6. (1) Using the idea of Observation 1.8, one can define the fundamental sequence 〈a0, a1, . . . , an〉 for anH fuzzy matroid (X, �) similarly (cf. [18]). We can prove that, for all a ∈ (0, a1), {�[a]|� ∈ �} = 2Y for some Y ∈ 2X

(this is a stronger result than Theorem 2.3 in [11]). However, it is still not known if (X, {�[a]|� ∈ �}) is a matroid fora ∈ [a1, 1].

(2) It is easy to construct a G–V fuzzy matroid (X, �) which has more than one fuzzy bases. Thus a G–V fuzzymatroid may not be an H fuzzy matroid by Proposition 5.5. An H fuzzy matroid may not be a G–V fuzzy matroid,either (see Example 5.9).

(3) It is easy to verify that (FI2) of Definition 5.2 is equivalent to (FI2) of Definition 5.4, so every H fuzzy matroidis a fuzzy matroid. But not vice versa clearly.

(4) There exists no implication between H fuzzy matroids and weak fuzzy matroids (see Examples 5.10 and 5.11).

Let (X, I) be a matroid and Y ⊆ X . If Y ∈ I, then we say that the degree of Y as an independent set is 1; if Y /∈ I,the degree is 0. Extending {0,1} to [0, 1] (or a completely distributive lattice L), a new approach to the fuzzification ofmatroids is proposed in [21].

Definition 5.7 (Shi [21, Definition 1]). Let X be a finite set. If a map � : 2X −→ [0, 1] (resp., L) satisfies the followingconditions:

(FI1) �(∅) = 1;(FI2) If Y, Z ∈ 2X and Y ⊇ Z , then �(Y )≤�(Z );(FI3) For any Y, Z ∈ 2X , if |Y | < |Z |, then there exists x ∈ Z − Y such that �(Y ∪ {x})≥�(Y ) ∧�(Z ),

then the pair (X, �) is called an S fuzzy matroid (resp., L-valued matroid).

Remark 5.8. (1) We know that an S fuzzy matroid is equivalent to a closed fuzzy pre-matroid (see [21, Remark 11]),so a closed G–V fuzzy matroid (i.e., a perfect hereditary fuzzy pre-matriod, see [19, Proposition 8]) is an S fuzzymatroid. An S fuzzy matroid is not a G–V fuzzy matroid in general (see Example 5.9).



(2) There is no implication between H fuzzy matroids (or fuzzy matroids) and S fuzzy matroids. It is easy toconstruct an S fuzzy matroid that is not an H fuzzy matroid (cf. [19, Example 1]). For an H fuzzy matroid that is not anS fuzzy matroid, see Example 5.9. Though S fuzzy matroids have some good properties (e.g., S fuzzy matroids can begeneralized to L-valued matroids easily), they have no hereditary property (see [21, Remark 12]), which is an essentialproperty of crisp matroids and possessed by other four kinds of fuzzy matroids in this section.

(3) It can easily be shown that the a-level structures of a weak fuzzy matroid are matroids using the same argumentsas in the proof of Theorem 2.1 in [3]. Thus, every closed weak fuzzy matroid is an S fuzzy matroid.

Example 5.9. Consider a two-element set X = {x, y}. For each m ∈ [0, 12 ], define �m ∈ [0, 1]X as follows:

�m(z) ={

12 + m, z = x,

12 − m, z = y.

Let �m = {�≤�m |� ∈ [0, 1]X } (m ∈ [0, 12 ]) and � = ⋃

m∈[0,1/2] �m . Then (X, �) is a fuzzy matroid, a weak fuzzymatroid, an H fuzzy matroid, and an S fuzzy matroid. But it is not a G–V fuzzy matroid. Let �∗ = �− {x1/2 ∨ y1/2},then (X, �∗) is still an H fuzzy matroid. Since (X, �∗) is not closed, it is not an S fuzzy matroid.

Example 5.10. Let X = {x, y, z} and � = {� ∈ [0, 1]X ||�|≤1}. It is easy to check that (X, �) is an H fuzzy matroid.Let � = x2/3 ∨ y1/3 and � = x1/3 ∨ y1/3 ∨ z1/3. Then |supp �| < |supp �| and m(�)≤m(�), but there exists no � whichsatisfies (FI2) of Definition 5.1. (X, �) is certainly not a G–V fuzzy matroid, either.

Example 5.11. Let X = {x, y, z} and � = {� ∈ [0, 1]X |�≤�1 or �≤�2}, where �1 = x1/2∨ y1/2 and �2 = y1/2∨ z1/2.It is easy to check that (X, �) is a G–V fuzzy matroid. Note that (X, �) has two fuzzy bases, by Proposition 5.5, (X, �)is neither an H fuzzy matroid nor a fuzzy matroid.

Remark 5.12. The relations between G–V fuzzy matroids, weak fuzzy matroids, fuzzy matroids, H fuzzy matroids andS fuzzy matroids (represented by G–V, W, F, H and S respectively) are illustrated as follows (where “←− • −→” means“there is no implication”, “−→” means “implies” but another direction “does not implies”; note that “G.V −→ S” and“W −→ S” are under the condition “closed”):

G-V

WH

F S

As mentioned above, there are two ways to generalize the notion of matroid and related combinatorial optimizationproblems. We now point out that there may be some connections between the two ways, which is an interesting subjectand will be worthy of consideration in future.

Let M = (X, I) be a matroid. The values of weights associated with element x ∈ X are from a given intervalWx = [w−x , w+x ], where w−x ≥0. In [2,13], some matroid-based optimization problems connected with interval weightedmatroid (X, I) are considered. We take DPosE (i.e. given an interval weighted matroid (X, I) and element x ∈ X ,decide whether x is possibly optimal) as an example. If the weights are given as intervals, the possibly optimal elementx has the following characterization: x ∈ X is possibly optimal if and only if x ∈ B(w−x ,x) [13, Theorem 1]. If x is



a possibly optimal element, then the degree of possibility that x is optimal is 1; otherwise the degree is 0. Extending{0, 1} to [0,1], the optimization problems are extended to the case in which the weights are given as fuzzy intervalsor fuzzy numbers. Then the degree of possibility that an element x is optimal is characterized by means of a familyof interval weighted matroids {Ma : a ∈ [0, 1]}. Note that there are common points between the extension frommatroids to S fuzzy matroids and the extension of the matroid-based optimization problem form interval weights tofuzzy interval weights. In addition, note some properties of Ma , for example, if a < b (a, b ∈ [0, 1]), then the conditionthat x is possibly optimal in Mb implies that x is possibly optimal in Ma . We suspect that there may generate some Sfuzzy matroid in the optimization problems. In [22], S fuzzy matroids are generalized to lattice-valued fuzzy case. Theproblems discussed in [2,13] may generalize to lattice-valued fuzzy case similarly.

6. Conclusions

In Section 5, we compare five kinds of fuzzy matroids. Although there may be other ways to fuzzify matroids, G–Vfuzzy matroids do preserve many of the basic properties of crisp matroids (see [3–10,14–16]). Since the set of G–Vfuzzy matroids contains crisp matroids as its one part, some properties that hold for crisp matroids may do not hold forG–V fuzzy matroids. To investigate these properties, maybe we could work with a certain class of G–V fuzzy matroids.In [6], a necessary and sufficient condition for a greedy algorithm to find a maximal fuzzy bases is established for aclosed regular G–V fuzzy matroid, and in [14], closure axioms are proposed for closed singular G–V fuzzy matroids.In this paper, to investigate the connectedness of G–V fuzzy matroids, a class of G–V fuzzy matroids (namely, refinedG–V fuzzy matroids) is introduced. It is shown that each refined G–V fuzzy matroid is regular and the transitivitytheorem holds for refined G–V fuzzy matroids. Based on the transitivity of fuzzy circuits, connectedness is naturallydefined for refined G–V fuzzy matroids, and some properties about connectedness are studied. These results enrich thetheory of G–V fuzzy matroids and may induce some better results.

Acknowledgments

The authors would like to thank very sincerely the unknown reviewers for their valuable comments and helpfulsuggestions.

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