Intuitionistic fuzzy matroids

Abstract

In this paper, we will define intuitionistic fuzzy matroid and study their properties. First, we analyse the two approaches to fuzzification of matroids and decide to use an indirect approach. After some preliminaries, we define an intuitionistic fuzzy matroid as a prefect intuitionistic fuzzy pre-matroid. Second, we study two pair of classical operators imposed on intuitionistic fuzzy matroids. Finally, we induce a G-V fuzzy matroid from a given intuitionistic fuzzy matroid and study the relationship of H fuzzy matroids and intuitionistic fuzzy matroids. Besides, we introduce an approach to construction of G-V fuzzy matroids and point out the connection of fuzzy matroids and fuzzy rough set models.

Keywords

Intuitionistic fuzzy sets matroids rough sets

1 Introduction

The concept of matroid was first introduced by Whitney [26] in 1935 to provide a unifying abstract treatment of linear algebra and graph theory. Matroids now have been proved to be essential important in many fields [25]. In recent years, matroids are also generalized and used as a tool to other mathematical branches, such as to rough sets [17], concept lattices [21] and fuzzy sets [8].

After Goetschel and Voxman’s first introduction of fuzzy matroids (named G-V fuzzy matroid) [8], they studied properties of G-V fuzzy matroids thoroughly in their subsequent papers [9 –12]. Since many properties of matroids can be generalized to G-V fuzzy matroids, it becomes in some degree a blueprint of all subsequent fuzzifications of matroids.

In 1993, Hsueh [13] proposed his fuzzy matroid (named H fuzzy matroid) by using a different cardinality of fuzzy sets with G-V fuzzy matroids. Shi [23] used the idea of degree to define S fuzzy matroid in 2010. AL-Haward [1] in 2012 generalized the closed set axioms of matroids to fuzzy setting and defined AH fuzzy matroids. On the one hand, we do not know which fuzzy matroid is better than others since there is no immediate criterion. It seems that these fuzzy matroids all perform well in some aspects and badly in other aspects. For example, although G-V fuzzy matroids have many good properties analogous to matroids, they do not necessarily have fuzzy bases (cf. [9]). H fuzzy matroids must possess fuzzy bases, while level-structures of H fuzzy matriods need not be crisp matroids (cf. [18]). S fuzzy matroids are closed fuzzy pre-matroids (i.e. special G-V fuzzy matroids, see [23]) and AH fuzzy matroids do not have the augmentation property of fuzzy base [1]. On the other hand, all these fuzzy matroids are defined by the direct approach. That is, these fuzzy matroids are all based on direct generalizations of axiomatic definition of matroid from crisp setting to fuzzy setting. Matroids have many different but equivalent axiomatic definitions, different fuzzy generalizations may not be equivalent and thus various fuzzy matroids are proposed (e.g. H fuzzy matroids come from the independent axiom and AH fuzzy matroids is from the closed set axiom). Even for one axiomatic definition, take the independent axiom as example (see Definition 2.1), different fuzzifications of (I2) lead to various fuzzy matroids (see, e.g. G-V fuzzy matroids and H fuzzy matroids). For relations of fuzzy matroids mentioned above, see [15].

After the introduction of the concept of fuzzy sets by Zadeh [31], several researches studied the generalizations of the notion of fuzzy set. The idea of intuitionistic fuzzy set was first published by Atanassov [2] and many work by the author and other researchers appeared subsequently (see [3]). The notion of intuitionistic fuzzy set is an important generalization of fuzzy sets, and some mathematical structures have been generalized to intuitionistic fuzzy setting, such as intuitionistic fuzzy topologies [5], intuitionistic fuzzy graphs [3], et al.

As stated previously, matroids have been generalized to fuzzy setting and properties of fuzzy matroids are obitained. In this paper, we want to generalize matroids to intuitionistic fuzzy setting and define the concept of the so-called intuitionistic fuzzy matroid. What we should mention here is that an indirect approach is used in this paper. One reason is that the direct generalization of fuzzy matroids to intuitionistic fuzzy matroids seems lack of innovation. Another reason is that the concept of matroid refers to the cardinality of a set and the cardinality of intuitionistic fuzzy sets may make the generalization of (I2) complicated.

Rough set theory is an important tool to deal with situations in which the objects of a given universe can be identified only within the limits of imprecise data by an indiscernibility relation. In rough set theory, we depict a vague concept by a pair of approximation operators. As shown by the authors in [16], the approximation operators are exactly the closure and inner operators of a matroid. Rough sets models have been generalized to fuzzy setting and successfully applied to many fields (e.g., see [4]). The relationship of fuzzy approximation operators and fuzzy closure operators of a fuzzy matroid has never been studied. Once our intuitionistic fuzzy matroids constructed, taking intuitionistic fuzzy closure operators as approximation operators, we can introduce intuitionistic fuzzy rough set models. In other words, our work in this paper can induce a new generalized rough set model based on intuitionistic fuzzy matroids and results in the paper may be viewed as theoretic foundations for future applications of this new model.

The rest of this paper is arranged as follows. In Section 2, we present basic notions and results of intuitionistic fuzzy sets and matroids. In Section 3, we introduce the notion of intuitionistic fuzzy matroid based on an indirect approach. Two pair of classical operators on intuitionistic fuzzy matroids are studied in Section 4. The connection of intuitionistic fuzzy matroids and G-V fuzzy matroids is investigated in Section 5. This paper is then concluded with a brief summary and an outlook for further research.

2 Preliminaries

In this section, we introduce the concepts related to intuitionistic fuzzy sets and matroids, which are mainly from [3] and [25] respectively.

2.1 Intuitionistic fuzzy sets

Definition 2.1. [3] Let X be a nonempty fixed set. An intuitionistic fuzzy set (IFS) A in X is defined as an object of the following form $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 ∣ x \in X},$ where the functions: $μ_{A} : X \to [0, 1]$ and $ν_{A} : X \to [0, 1]$ defined the degree of membership and the degree of non-membership of the element x ∈ X respectively, and for every x ∈ X: $0 \leq μ_{A} (x) + ν_{A} (x) \leq 1 .$

The set of all intuitionistic fuzzy sets in X will be denoted by X^∗. Obviously, each ordinary fuzzy set may be written as ${〈 x, μ_{A} (x), 1 - μ_{A} (x) 〉 ∣ x \in X} .$

The set of all fuzzy set are denoted by [0, 1] ^X. Usually, we use μ_A to denote the fuzzy set for short, or just μ if there is no confusion.

Definition 2.2. [3] For A, B ∈ X^∗, the inclusion relation is defined by A ⊆ B iff μ_A (x) ≤ μ_B (x) & ν_A (x) ≥ ν_B (x) (∀ x ∈ X).

In the following, we introduce the notion of cuts of intuitionistic fuzzy sets.

Definition 2.3. [3] Let A ∈ X^∗. For any ordered pair <α, β>, where 0 ≤ α+ β ≤ 1, define $A_{< α, β >} = {x \in X ∣ μ_{A} (x) \geq α, ν_{A} (x) \leq β},$ we call it the <α, β>-cut of A.

The <α, 1 - α>-cut of a fuzzy set μ is usually denoted by μ_[α] (i.e. μ_[α] = {x ∈ X ∣ μ (x) ≥ α}). Let τ be a family of intuitionistic fuzzy sets in X (i.e. τ ⊆ X^∗), we use τ_<α,β> to denote the set {A_<α,β> ∣ A ∈ τ}.

2.2 Matroids and fuzzy matroids

Definition 2.4. [25] Let X be a finite set and $I$ be a nonempty subset of 2^X (the set of all subsets of X), then $(X, I)$ is a set system. A set system $(X, I)$ is called a matroid if the following conditions hold:

If $Y \in I,$ and Z ⊆ Y, then $Z \in I$ .

Let $Y, Z \in I$ , and |Z| < |Y| (where |Y| denotes the cardinality of Y) , then there exists a set $W \in I$ such that Z ⊂ W ⊆ Z ∪ Y .

Let $M = (X, I)$ be a matroid. The members of $I$ are the independent sets of M. A set in $I$ is maximal in the sense of inclusion is called a base of the matroid M. Let $M = (U, I)$ be a matroid. The elements of $I$ are called independent sets of M. We often write $I (M)$ for $I$ , particularly when several matroids are being considered. The maximal element in $I$ in the sense of inclusion is called a base of the matroid M. If A ⊆ U and $\notin I$ , then A is a dependent set. A minimal dependent subset of U (in the sense of inclusion) is called a circuit of M. The function ρ : 2^U → N, defined by $ρ (Y) = max {| X | ∣ X \subseteq Y, X \in I} (Y \subseteq U)$ is called a rank function of the matroid. For every X ⊆ U, let cl_M (X) = {a ∈ U ∣ ρ (X) = ρ (X ∪ {a})}, we call it the closure of X in (U, $I$ ). When there is no confusion, we use the symbol cl (X) for short.

In Whitney’s founding paper of matroids [26], two fundamental classes of matroids are considered. One arises from matrices and another is from graphs. As a particular example of the class of matroids from graphs, consider the following.

Example 2.5. Let E be the set of edges of a graph G and let $X \in I$ (⊆2^E) if and only if X does not contain a cycle of G 1 . Then it is easy to check that $I$ is the collection of independent sets of a matroid on E, called the cycle matroid of G and denoted by M (G). Consider the three graphs shown in Fig. 1. Then the cycle matroid is $M (G) = (E, I$ ), where E = {1, 2, 3, 4, 5}, and $I = 2^{{1, 2, 3}} \cup 2^{{1, 3, 4}}$ .

Fig.1

A graph G.

In 1988, Goetschel and Voxman introduced the notion of fuzzy matroids (G-V fuzzy matroids) as follows.

Definition 2.6. [8] Let X be a nonempty finite set and Ψ ⊆ [0, 1] ^X. If Ψ satisfies

(FI1) If μ, ν ∈ Ψ, μ ≤ ν and ν ∈ Ψ, then μ ∈ Ψ.

(GVFI2) If μ, ν ∈ Ψ and 0 < |supp μ| < |supp ν|, then there exists ω ∈ Ψ such that

μ < ω ≤ μ ∨ ν .

m (ω) ≥ min {m (μ) , m (ν)} .

then we call (X, Ψ) a G-V fuzzy matroid.

3 Intuitionistic fuzzy matroids

In this section, we first introduce an indirect approach to the definition of intuitionistic fuzzy matroid and conclude that an intuitionistic fuzzy mat-roid should be an hereditary intuitionistic fuzzy pre-matroid. Then we define an equivalence relation based on level-structure and consider the greatest ele-ment of the equivalence class as an intuitionistic fuzzy matroid. Finally, we study the relations of some conditions proposed on intuitionistic fuzzy set systems.

3.1 An indirect approach

As pointed out in Section 1, there are two approaches to generalizing matroids to intuitionistic fuzzy matroids. One is the direct approach. That is, we should generalize (I1) and (I2) in Definition 2.4 to the intuitionistic fuzzy case. The other is an indirect approach and we will consider it later.

We first give the definition of intuitionistic fuzzy set system (IFSS).

Definition 3.1. Let τ be a family of IFSs in X (i.e. τ ⊆ X^∗). Then we call (X, τ) an IFSS.

Based on Definition 3.1, (I1) can be easily generalized to

(IFI1) ∀A, B ∈ X^∗, A ∈ τ and B ⊆ A imply B ∈ τ.

Unfortunately, when we generalize (I2) to (IFI2), there are many different ways. And we do not know which approach is the best and thus intuitionistic fuzzafication of (I2) is hard to process. For example, (I2) refers to the cardinality of a set. While there are many kinds of cardinalities of IFS (see [6]) and every cardinality can correspond to a special form of (IFI2).

In fact, when generalizing matroids to fuzzy setting, the same question also arises. In 1988, Goetschel and Voxman first generalized Definition 2.4 and defined a kind of fuzzy matroid (G-V fuzzy matroid); Hsueh in 1993 introduced his fuzzy matroid (H fuzzy matroid). The difference between G-V fuzzy matroids and H fuzzy matroids lies in various generalizations of (I2) 2 . All these fuzzy matroids are defined by the direct approach.

Remark 3.2. Based on discussion previously, we can see that the direct approach leads to different intuitionistic fuzzy matroids and we can not evaluate which generalization is better than others. In the following, we use an indirect approach. That is, we temporarily do not consider the question of generalization of (I1) and (I2), but first focus on the following question:

What is the fundamental property that should be satisfied by an intuitionistic fuzzy matroid?

Remark 3.3. There is no immediate criterion which kind of intuitionistic fuzzy matroid is more “natural” than others. Nevertheless, it seems to be widely accepted that any intuitionistic fuzzyfied structure has to have an analogous crisp structures as its levels. Consequently, an IFSS (X, τ) pretending to be an intuitionistic fuzzyfication of a crisp matroid should have crisp matroids as its levels.

i.e., (X, τ_<α,β >) should be a matroid for each α, β ∈ [0, 1], where τ_<α,β> = {A_<α,β> ∣ A ∈ τ} .

Note that for every β ∈ [1 - α, 1], we have A_<α,β> = A_<α,1-α> . Thus when we consider the levels of an intuitionistic fuzzy set system τ_<α,β>, we tacitly admit α + β ≤ 1. We now decide to postpone our discussion of level-structure until later. As pointed out previously, (I1) could be easily generalized to intuitionistic fuzzy setting. Let (X, τ) be an IFSS. Then we call (X, τ) an hereditary IFSS if it satisfies the following condition:

(IFI1) ∀A, B ∈ X^∗, A ∈ τ and B ⊆ A imply B ∈ τ.

An hereditary IFSS has the following property.

Proposition 3.4. Let (X, τ) be an hereditary IFSS. Then for every pair of α and β (∈ [0, 1]) , (X, τ_<α,β>) satisfies (I1).

Proof. Suppose that Y ∈ τ_<α,β> and Z ⊆ Y, we need to prove Z ∈ τ_<α,β>. Since Y ∈ τ_<α,β>, there exists A ∈ τ such that A_<α,β> = Y. That is, ∀y ∈ Y, μ_A (y) ≥ α and ν_A (y) ≤ β. We now construct an intuitionistic fuzzy set B as follows. ${\begin{matrix} μ_{B} (x) = μ_{A} (x) \land ν_{B} (x) = ν_{A} (x), & x \in Z; \\ μ_{B} (x) = 0 \land ν_{B} (x) = 1, & x \in X - Z . \end{matrix}$

Note Z ⊆ Y, and we can easily check B_<α,β> = Z. Obviously, B ⊆ A. It follows from (IFI1) that B ∈ τ. Thus Z ∈ τ_<α,β>.

The converse of Proposition 3.4 does not necessarily hold (cf. Example 3.7). We now continue to our discussion of level-structure of an IFSS.

Definition 3.5. Let (X, τ) be an IFSS. For every pair of α and β (∈ [0, 1]), if (X, τ_<α,β >) is a matroid, then we call (X, τ) an intuitionistic fuzzy pre-matroid.

Remark 3.6. Based on our analysis in Remark 3.3, an intuitionistic fuzzy matroid should be an intuitionistic fuzzy pre-matroid. Besides, (IFI1) is a generalization of (I1). Thus an intuitionistic fuzzy matroid should be an hereditary intuitionistic fuzzy pre-matroid.

At last of this subsection, we propose two intuitionistic fuzzy pre-matroids. One is hereditary, and another is not.

Example 3.7. Let X = {a, b}, define three intuitionistic fuzzy sets as follows: $\begin{matrix} A & = & {< a, 0.5, 0.5 >, < b, 0.5, 0 >}, \\ B & = & {< a, 0.5, 0.5 >, < b, 0, 1 >}, \\ C & = & {< a, 0, 1 >, < b, 0.5, 0 >} . \end{matrix}$

Again, let τ = ↓ A = {D ∣ D ⊆ A} and ι = {A, B, C}. It is easy to check 3 $τ_{< α, β >} = {\begin{matrix} 2^{{a, b}}, & α = 0.5, β = 0.5, \\ {\emptyset, {b}}, & α = 0.5, β < 0.5, \\ 2^{{a, b}}, & α < 0.5, β \geq 0.5, \\ {\emptyset, {b}}, & α < 0.5, β < 0.5, \end{matrix}$ and τ_<α,β> = ι_<α,β> . Thus (X, τ) and (X, ι) are intuitionistic fuzzy pre-matroids, and further (X, τ) is hereditary.

3.2 Intuitionistic fuzzy matroids

In Example 3,7, though ι ⊂ τ, the two related IFSSs (X, τ) and (X, ι) have the same level-structure (i.e. τ_<α,β> = ι_<α,β>). In fact, there are many IFSSs satisfying this property. For example, for every E ∈ τ - ι, (X, ι ∪ {E}) has the same level-structure with (X, τ) (we say that they are isolevel). We now define this relation (denoted by ∼) formally as follows. $(X, τ_{1}) \sim (X, τ_{2}) if and only if τ_{1 < α, β >} = τ_{2 < α, β >}$

That is, (X, τ₁) ∼ (X, τ₂) if and only if they are isolevel. Obviously, ∼ is an equivalence relation and we use [(X, τ)] to denote the equivalent class containing (X, τ). We often omit specific mention of X and use the notation [τ] if no confusion will arise. The set [τ] ordered by inclusion forms a poset ([τ] , ≤), which has the following properties.

Proposition 3.8. (1) For any τ₁, τ₂ ∈ [τ], we have τ₁ ∨ τ₂ ∈ [τ], but τ₁ ∧ τ₂ does not necessarily belong to [τ].

(2) The greatest element of ([τ] , ≤) exists, and we denote it by ⊤_τ, or simply by ⊤ if there is no confusion. The least element of ([τ] , ≤) does not necessarily exist.

(3) ⊤ is the greatest element of ([τ] , ≤) if and only if $\begin{matrix} A \in X^{*} and A_{< α, β >} \in ⊤_{< α, β >} \\ (\forall α, β \in [0, 1]) implies A \in ⊤ . (★) \end{matrix}$

Proof. For the related counterexamples to (1) and (2), see Example 3.9.

It follows easily from the definition of [τ].

∀α, β ∈ [0, 1], ι ∈ [τ], we have ι_<α,β> = τ_<α,β>. Thus $⋃_{κ \in [τ]} κ_{< α, β >} = (⋃_{κ \in [τ]} κ)_{< α, β >} = τ_{< α, β >} .$

That is, $⊤ = ⋃_{κ \in [τ]} κ \in [τ] .$

Suppose A ∈ X^∗ and A_<α,β> ∈ τ_<α,β> (∀α, β ∈ [0, 1]). Note τ⊆ ⊤, thus τ_<α,β> ⊆ ⊤ _<α,β>. It follows from A_<α,β> ∈ ⊤ _<α,β> that $(⊤ \cup {A})_{< α, β >} = ⊤_{< α, β >} = τ_{< α, β >} .$

By the definition of [τ], we have ⊤ ∪ {A} ∈ [τ]. Since ⊤ is the greatest element of [τ], we have ⊤∪ {A} = ⊤. That is, A∈ ⊤.

Conversely, suppose that (★) holds and ⊤^∗ is the greatest element of [τ]. Then ⊤ ⊆ ⊤ ^∗. If ⊤^∗ -⊤ ≠ ∅, let A∈ ⊤ ^∗ - ⊤. Since A ∈ ⊤ ^∗, we have $A_{< α, β >} \in ⊤_{< α, β >}^{*} = τ_{< α, β >} (\forall α, β \in [0, 1])$ . It holds from (★) that A∈ ⊤. It is a contradiction, and thus ⊤^∗ =⊤. This concludes the proof.

Example 3.9. Let X = {a, b}. Define five intuitionistic fuzzy sets as follows: $\begin{matrix} A & = & {< a, 1, 0 >, < b, 0, 1 >}, \\ B & = & {< a, 0.5, 0.5 >, < b, 0.5, 0.5 >}, \\ C & = & {< a, 0.5, 0.5 >, < b, 0, 1 >}, \\ D & = & {< a, 0, 1 >, < b, 0.5, 0.5 >}, \\ E & = & {< a, 1, 0 >, < b, 0.5, 0.5 >} . \end{matrix}$

Let τ = {A, B, C, D} and ι = {C, D, E} . It is easy to check that τ and ι are isolevel and they are minimal elements of ([τ] , ≤). Thus ([τ] , ≤) has no least element. Further, τ ∧ ι = {C, D} ∉ [τ] .

Remark 3.10. (1) From Proposition 3.8, we can see that ([τ] , ≤) forms a complete join-semilattice.

(2) Let (X, τ) be an IFSS, if τ satisfies (★), we say that (X, τ) is perfect. As pointed out in Remark 3.6, an intuitionistic fuzzy matroid should be an hereditary intuitionistic fuzzy pre-matroid. For an intuitionistic fuzzy pre-matroid (X, τ), it need not be hereditary (see, e.g. (X, ι) in Example 3.7). As discussed above, [τ] is a complete join-semilattice and the greatest element of ([τ] , ≤) is perfect. We will prove in the following that an perfect intuitionistic fuzzy pre-matroid is hereditary.

Proposition 3.11. A perfect intuitionistic fuzzy pre-matroid is hereditary.

Proof. Let (X, τ) be a perfect intuitionistic fuzzy pre-matroid. ∀A ∈ τ and B ⊆ A, we have $B_{< α, β >} \subseteq A_{< α, β >} \in τ_{< α, β >} (\forall α, β \in [0, 1]) .$

Note (X, τ) is an intuitionistic fuzzy pre-matroid, it follows from (I1) that $B_{< α, β >} \in τ_{< α, β >} (\forall α, β \in [0, 1]) .$

Since (X, τ) is perfect, we have B ∈ τ, which concludes the proof.

Remark 3.12. The converse of Proposition 3.11 need not hold. That is, a hereditary intuitionistic fuzzy pre-matroid need not be perfect.

Example 3.13. Let X = {a, b}. Define two intuitionistic fuzzy sets as follows: $\begin{matrix} A & = & {< a, 1, 0 >, < b, 0, 1 >}, \\ B & = & {< a, \frac{1}{2}, \frac{1}{2} >, < b, 1, 0 >} . \end{matrix}$

Let τ = ↓ A ∪ ↓ B, then (X, τ) is obviously a hereditary IFSS. It is easy to check that $τ_{< α, β >} = {\begin{matrix} {\emptyset, {a}, {b}}, & α \in (0, \frac{1}{2}], β \in (0, 1 - α], \\ 2^{{a, b}}, & α \in (0, \frac{1}{2}], β \in (0, 1 - α] . \end{matrix}$

Thus (X, τ) is an intuitionistic fuzzy pre-matroid. Let $A = {< a, 1, 0 >, < b, \frac{1}{2}, \frac{1}{2} >},$ then it is easy to verify $A_{< α, β >} \in τ_{< α, β >} (\forall α, β \in (0, 1]) .$

But A ∉ τ, which show that (X, τ) is not perfect.

Based on Remark 3.6 and Remark 3.10, we define Definition 3.14. A perfect intuitionistic fuzzy pre-matroid is called an intuitionistic fuzzy matroid.

For an intuitionistic fuzzy matroid (X, τ), we list its main properties mentioned in this section as follows. And more properties will be given in the subsequent sections.

(X, τ_<α,β>) is a matroid (∀ α, β ∈ [0, 1]) .

For every A ∈ X^∗, A_<α,β> ∈ τ_<α,β> (∀α, β ∈ [0, 1]) implies A ∈ τ.

If ι ∈ [τ], then ι ⊆ τ.

4 Operators on intuitionistic fuzzy matroids

After IFSs were introduced, plenty of operators on IFSs were proposed and studied (see [3]). In this section, we discuss two important pairs of operators imposed on intuitionistic fuzzy matroids.

1. “Necessity” and “Possibility” operators.

Definition 4.1. [3] For every IFSA, we define the necessary operator $□ A = {< x, μ_{A} (x), 1 - μ_{A} (x) >},$ the possibility operator $◊ A = {< x, μ_{A} (x), 1 - μ_{A} (x) >} .$

For an IFSS (X, τ), we now consider the impact of the necessary operator □ on (X, τ). As for the possibility operator ◊, we can discuss the impact similarly. First, we define $□ τ = {□ A ∣ A \in τ} .$

In [5], Coker defined intuitionistic fuzzy topological spaces and pointed out that the necessary operator and the possibility operator could induce new intuitionistic fuzzy topological spaces from a given one. For an intuitionistic fuzzy matroid (X, τ), we want to know whether (X, □ τ) is an intuitionistic fuzzy matroid.

Example 4.1. Let X = {a, b}. Define an intuitionistic fuzzy sets as follows: $A = {< a, \frac{1}{2}, \frac{1}{4} >, < b, \frac{1}{4}, \frac{1}{4} >},$

Obviously, (X, ↓ A) is an intuitionistic fuzzy matroid. It is easy to verify that $□ τ_{< α, β >} = {\begin{matrix} {\emptyset}, & α \in (\frac{1}{2}, 1] . \\ {{a}, \emptyset}, & α \in (\frac{1}{4}, \frac{1}{2}], β \in [\frac{1}{2}, 1), \\ {\emptyset}, & α \in (\frac{1}{4}, \frac{1}{2}], β \in (0, \frac{1}{2}) . \\ 2^{{a, b}}, & α \in (0, \frac{1}{4}], β \in [\frac{3}{4}, 1) . \\ {{a}, \emptyset}, & α \in (0, \frac{1}{4}], β \in [\frac{1}{2}, \frac{3}{4}) . \\ {\emptyset}, & α \in (0, \frac{1}{4}], β \in (0, \frac{1}{2}) . \end{matrix}$

Now we consider the intuitionistic fuzzy set $B = {< a, \frac{1}{4}, \frac{1}{2} >, < b, 0, 1 >}$ . It is easy to check $A_{< α, β >} = {\begin{matrix} {\emptyset}, & α \in (\frac{1}{4}, 1] . \\ {a}, & α \in (0, \frac{1}{4}], β \in [\frac{1}{2}, 1) . \\ {\emptyset}, & α \in (0, \frac{1}{4}], β \in (0, \frac{1}{2}) . \end{matrix}$

Thus A_<α,β> ∈ □ τ_<α,β> (∀ α, β ∈ (0, 1]). But A ∉ □ τ, which shows that (X, □ τ) is not perfect, so it is not an intuitionistic fuzzy matroid.

Remark 4.2. For an intuitionistic fuzzy matroid (X, τ), □τ actually is a family of fuzzy sets. In Example 4.1, A is not a fuzzy set. In fact, if a fuzzy set C satisfies $C_{< α, β >} \in □ τ_{< α, β >} (\forall α, β \in (0, 1]),$ we can prove C ∈ □ τ. Besides, we take (X, □ τ) as an IFSS and investigate its <α, β>-cut □τ_<α,β> (∀ α, β ∈ (0, 1]) . While for a fuzzy set system (X, □ τ), it seems reasonable to consider only the cut of fuzzy sets (i.e. □τ_<α,1-α> (∀ α ∈ (0, 1]) . We will prove in the next section that (X, □ τ) possesses good properties in the sense of fuzzy set systems.

2. Operator G_a,b.

Definition 4.3. [3] For every IFSA, we define the operator $G_{a, b} (A) = {< x, a μ_{A} (x), b ν_{A} (x) > ∣ x \in X} .$

For an IFSS (X, τ), we define $τ^{< a, b >} = {G_{a, b} (A) ∣ A \in τ} (\forall a, b \in [0, 1]) .$

In the following, we will study the properties of (X, τ^<a,b>).

Lemma 4.4. Let (X, τ) be an IFSS, then

∀A ∈ τ, A_<α,β> = G_a,b (A) _<aα,bβ> (∀ a, b, α, β ∈ [0, 1]) .

∀A ∈ τ^<a,b>, then ∀a, b ∈ (0, 1] and α, β ∈ [0, 1] , we have

$G_{\frac{1}{a}, \frac{1}{b}} (A) \in τ and G_{a, b} (G_{\frac{1}{a}, \frac{1}{b}} (A)) = A .$

Proof. (1) Obviously, $\begin{matrix} x \in A_{< α, β >} & \Leftrightarrow & μ_{A} (x) \geq α and ν_{A} (x) \leq β \\ \Leftrightarrow & a μ_{A} (x) \geq a α and b ν_{A} (x) \leq b β \\ \Leftrightarrow & x \in G_{a, b} (A)_{< a α, b β >}, \end{matrix}$ thus A_<α,β> = G_a,b (A) _<aα,bβ> (∀ a, b, α, β ∈ [0, 1]).

(2) Since A ∈ τ^<a,b>, there exist B ∈ τ such that G_a,b (B) = A . That is aμ_B (x) = μ_A (x) and bν_B (x) = ν_A (x) (∀ x ∈ X). So $μ_{B} (x) = \frac{1}{a} μ_{A} (x)$ and $ν_{B} (x) = \frac{1}{a} ν_{A} (x)$ , that is $G_{\frac{1}{a}, \frac{1}{b}} (A) = B \in τ .$ Note that we also have checked that $G_{a, b} (G_{\frac{1}{a}, \frac{1}{b}} (A)) = A$ , which concludes the proof.

Remark 4.5. In Lemma 4.4, note a, b ∈ (0, 1], thus $\frac{1}{a}, \frac{1}{b} \geq 1$ . The operator $G_{\frac{1}{a}, \frac{1}{b}} (A)$ seems contrary to Definition 4.3 in which a and b are not bigger than 1. In the proof, from A ∈ τ^<a,b>, we can imply that ${< x, \frac{1}{a} μ_{A} (x), \frac{1}{a} ν_{B} (x) > ∣ x \in X}$ is well-defined. For simplicity, we abuse notation by $G_{\frac{1}{a}, \frac{1}{b}} (A)$ for this operation.

Lemma 4.6. If (X, τ) is an intuitionistic fuzzy matroid, then (X, τ^<a,b>)(∀ a, b ∈ [0, 1]) is hereditary.

Proof. Let A ∈ τ^<a,b>, B ∈ X^∗ and B ⊆ A . It is easy to check that $G_{\frac{1}{a}, \frac{1}{b}} (B) \subseteq G_{\frac{1}{a}, \frac{1}{b}} (A) .$

By Lemma 4.4, we have $G_{\frac{1}{a}, \frac{1}{b}} (A) \in τ .$ Since (X, τ) is hereditary, we have $G_{\frac{1}{a}, \frac{1}{b}} (B) \in τ$ . It follows from Lemma 4.4 again that $B = G_{a, b} (G_{\frac{1}{a}, \frac{1}{b}} (B)) \in τ^{< a, b >}$ , which completes the proof.

Proposition 4.7. Let (X, τ) be an intuitionistic fuzzy matroid, then (X, τ^<a,b>) is also an intuitionistic fuzzy matroid(∀a, b ∈ [0, 1]).

Proof. First, we show that (X, τ^<a,b>) is an intuitionistic fuzzy pre-matroid. For every pair of α and β (α, β ∈ [0, 1]), we need to show that $(X, τ_{< α, β >}^{< a, b >}$ ) is a matroid. By Lemma 4.6, (X, τ^<a,b>) is hereditary. It follows from Proposition 3.4 that $(X, τ_{< α, β >}^{< a, b >}$ ) satisfies (I1). To show that $(X, τ_{< α, β >}^{< a, b >}$ ) satisfies (I2), suppose $Y, Z \in τ_{< α, β >}^{< a, b >}$ and |Y| < |Z|. By the definition of τ^<a,b>, there exist A¹, A² ∈ τ^<a,b> such that $A_{< α, β >}^{1} = Y$ and $A_{< α, β >}^{2} = Z$ . By Lemma 4.4(2), $G_{\frac{1}{a}, \frac{1}{b}} (A^{1}), G_{\frac{1}{a}, \frac{1}{b}} (A^{2}) \in τ .$

It holds from Lemma 4.4(1) and (2) that $\begin{matrix} G_{\frac{1}{a}, \frac{1}{b}} (A^{1})_{< \frac{α}{a}, \frac{β}{b} >} & = & G_{a, b} (G_{\frac{1}{a}, \frac{1}{b}} (A^{1}))_{< a . \frac{α}{a}, b . \frac{β}{a} >} \\ = & A_{< α, β >}^{1} = X, \\ G_{\frac{1}{a}, \frac{1}{b}} (A^{2})_{< \frac{α}{a}, \frac{β}{b} >} & = & G_{a, b} (G_{\frac{1}{a}, \frac{1}{b}} (A^{2}))_{< a . \frac{α}{a}, b . \frac{β}{a} >} \\ = & A_{< α, β >}^{1} = Y . \end{matrix}$

Since (X, τ) is an intuitionistic fuzzy pre-matroid, $(X, τ_{< \frac{α}{a}, \frac{β}{b} >})$ is a matroid. It holds from (I2) that there is $M \in τ_{< \frac{α}{a}, \frac{β}{b} >}$ such that Y ⊆ M ⊆ Y ∪ Z . By the definition of $τ_{< \frac{α}{a}, \frac{β}{b} >}$ , there exists B ∈ τ such that $B_{< \frac{α}{a}, \frac{β}{b} >} = M$ . By Lemma 4.4(1), $B_{< \frac{α}{a}, \frac{β}{b} >} = G_{a, b} (B)_{< a . \frac{α}{a}, b . \frac{β}{b} >} = G_{a, b} (B)_{< α, β >} = M .$

Note G_a,b (B) ∈ τ^<a,b>, thus $M \in τ_{< α, β >}^{< a, b >}$ , which verifies (I2). Therefore, $(X, τ_{< α, β >}^{< a, b >}$ ) is a matroid.

Now we show that (X, τ^<a,b>) is perfect. Let A ∈ X^∗ and $A_{< α, β >} \in τ_{< α, β >}^{< a, b >}$ (∀ α, β ∈ [0, 1]) . That is, $A_{< α, β >} = {x | μ_{A} (x) \geq α and ν_{A} (x) \leq β} \in τ_{< α, β >}^{< a, b >} .$

Obviously, x ∈ A_<α,β> if and only if $\frac{1}{a} μ_{A} (x) \geq \frac{1}{a} α and \frac{1}{b} ν_{A} (x) \leq \frac{1}{b} β .$

It holds from Definition 4.3 that $x \in G_{\frac{1}{a}, \frac{1}{b}} (A)_{< \frac{1}{a} α, \frac{1}{b} β >}$ . By Lemma 4.4, $G_{\frac{1}{a}, \frac{1}{b}} (A) \in τ .$ Again, it follows from Lemma 4.4 that $A = G_{a, b} (G_{\frac{1}{a}, \frac{1}{b}} (A)) \in τ^{< a, b >},$ which verifies perfectness and completes the proof.

5 Connections with fuzzy matroids

In this section, we discuss the relation of intuitionistic fuzzy matroids and fuzzy matroids. First we study the relationship of intuitionistic fuzzy matroids and G-V fuzzy matroids. And then we introduce two interesting topics on fuzzy matroids, which will be considered in intuitionistic fuzzy setting in future.

5.1 The relationship of intuitionistic fuzzy matroids and G-V fuzzy matroids

Let (X, τ) be an intuitionistic fuzzy matroid, we have pointed out that (X, □ τ) need not be an intuitionistic fuzzy matroid. Note that □τ actually is a family of fuzzy sets defined on X. We now prove that (X, □ τ) is a G-V fuzzy matroid. In the following, we use the membership function μ_A of an intuitionistic fuzzy set A to denote □A = {< x, μ_A (x) , 1 - μ_A (x) > ∣ x ∈ X}, or μ for short if there is no confusion.

Lemma 5.1. Let (X, τ) be an intuitionistic fuzzy matroid,

(X, □ τ) is hereditary.

∀r ∈ (0, 1], (X, □ τ_[r]) is a matroid, where $\begin{matrix} □ τ_{[r]} & = & {(μ_{A})_{[r]} ∣ A \in τ} and \\ (μ_{A})_{[r]} & = & {x ∣ μ_{A} (x) \geq r, x \in X} . \end{matrix}$

∀r ∈ (0, 1], (μ_A)_[r] ∈ □ τ_[r] implies μ_A ∈ □ τ.

Proof. (1) is trivial. (2) holds easily from the definition of intuitionistic fuzzy matroid and the following equation $□ τ_{[r]} = {(μ_{A})_{[r]} ∣ A \in τ} = τ_{< r, 1 - r >} .$

∀r ∈ (0, 1], (μ_A) _[r] ∈ □ τ_[r]. That is, there exists B^r ∈ τ such that $(μ_{A})_{[r]} = □ B_{< r, 1 - r >}^{r} = □ B_{[r]}^{r} .$

Define a fuzzy set C : μ_C (x) = sup ${r ∣ x \in B_{[r]}^{r}, r \in (0, 1]}$ . Obviously, $C_{< α, β >} = {\begin{matrix} □ B_{[α]}^{α} \in □ τ_{[α]}, & α = 1 - β . \\ {\emptyset}, & 0 < β < 1 - α . \end{matrix}$

Since □τ_[α] ⊆ τ_<α,1-α>, we have C_<α,β> ∈ τ_<α,β> (∀α, β ∈ (0, 1] and α + β ≤ 1). Because (X, τ) is perfect, C ∈ τ holds. Note C is a fuzzy set, C ∈ □ τ also holds. By (1), μ_A ⊆ C implies μ_A ∈ □ τ, which completes the proof of (3).

Proposition 5.2. Let (X, τ) be an intuitionistic fuzzy matroid. Then (X, □ τ) is a G-V fuzzy matroid.

Proof. (FI1) holds from Lemma 5.1(1). We show that (GVFI2) also holds. Suppose μ, ν ∈ □ τ and | supp μ| < |supp ν| . Let r = min {m (μ) , m (ν)}. Define two fuzzy sets on X as follows $μ_{1} = {\begin{matrix} r, & x \in supp μ, \\ 0, & otherwise, \end{matrix} ν_{1} = {\begin{matrix} r, & x \in supp ν, \\ 0, & otherwise . \end{matrix}$

Since μ₁ ≤ μ and ν₁ ≤ ν, by Lemma 5.1(1), we have μ₁, ν₁ ∈ □ τ. Note μ_1[r] = supp μ and ν_1[r] = supp ν. Thus |μ_1[r]| < |ν_1[r]|. It holds from Lemma 5.1(2) that (X, □ τ_[r]) is a matroid. By (I2), there is C ∈ □ τ_[r] such that μ_1[r] ⊂ C ⊆ ν_1[r] ∪ ν_1[r] . Define $ω = {\begin{matrix} μ (x), & x \in supp μ, \\ r, & x \in C - supp μ, \\ 0, & otherwise, \end{matrix}$

We now prove ω ∈ □ τ. For every a ∈ (0, r], $ω_{[a]} \subseteq ω_{[r]} = C \in □ τ_{[r]} \subseteq □ τ_{[a]} .$

By (I1), ω_[a] ∈ □ τ_[a]. If a > r, $ω_{[a]} = μ_{[a]} \in □ τ_{[a]} .$

We have verified that μ_[a] ∈ □ τ_[a] (∀a ∈ (0, 1]). So Lemma 5.1(3) implies that ω ∈ □ τ.

Obviously, μ < ω ≤ μ ∨ ν and m(ω) = r = min {m (μ) , m (ν)}, as desired. Therefore, (X, □ τ) is a G-V fuzzy matroid.

Remark 5.3. Note that an intuitionistic fuzzy matroid (X, τ) is hereditary (Proposition 3.11), thus it is easy to check that □τ = {A∈ τ ∣ A is a fuzzy set}. Thus for an intuitionistic fuzzy matroid (X, τ), Proposition 5.2 says that the family of fuzzy sets in τ forms a G-V fuzzy matroid.

Remark 5.4. For a G-V fuzzy matroid (X, τ), it is easy to check that (X, τ) need not be a hereditary intuitionistic fuzzy set system. Thus (X, τ) is not necessarily an intuitionistic fuzzy matroid. Let $\begin{matrix} ↓ τ = {A \in X^{*} ∣ \exists μ \in τ, \\ s . t . μ_{A} \leq μ and ν_{A} \geq 1 - μ}, \end{matrix}$ then (X, ↓ τ) is hereditary. However, we do not know whether (X, ↓ τ) is an intuitionistic fuzzy matroid.

Remark 5.5. Hsueh [13] in 1993 proposed his fuzzy matroids (named H fuzzy matroids). Li et al. [15] compared some kinds of fuzzy matroids and proposed an open problem that whether an H fuzzy matroid was a fuzzy pre-matroid. In [18], we constructed an counterexample to give this problem a negative answer. That is, every cut (level-structure) of an H fuzzy matroid need not be a matroid. Thus there is no connection between H fuzzy matroids and intuitionistic fuzzy matroids.

5.2 Constructions and approximations

In Example 2.5, we constructed a matroid from a given graph. By the famous representation theorem of fuzzy sets, we can induce a G-V fuzzy matroid from a given fuzzy graph (V, μ, ρ), where V is a nonempty set together with a pair functions μ : V → [0, 1] and ρ : V² → [0, 1] such that ∀x, y ∈ V, ρ (x, y) ≤ ρ (x) ∧ ρ (y) [22].

Let (V, μ, ρ) be a fuzzy graph, for every r ∈ (0, 1], consider the sequence of crisp graphes: (μ_[r], ρ_[r]). Obviously, we have $μ_{[a]} \subseteq μ_{[b]} and ρ_{[a]} \subseteq ρ_{[b]}, \forall a, b \in (0, 1], a > b .$

The sequence of crisp graphs can induce a sequence of matroids and we can construct fuzzy sets by the sequence of matroids. Take the fuzzy graph in Fig. 2 as an example, we illustrate how to construct a G-V fuzzy matroids.

Fig.2

A fuzzy graph G.

The fuzzy graph in Fig. 2 induces a sequence of crisp graphs by its cuts. The related cycle matroids with respect to these three graphs are $(V, I_{0.3})$ , $(V, I_{0.4})$ and $(V, I_{0.6})$ , where V = {1, 2, 3, 4}and $\begin{matrix} I_{0.6} & = & {\emptyset, {1}} \\ I_{0.4} & = & {\emptyset, {1}, {2}, {4}, {1, 2}, {1, 4}} \\ I_{0.3} & = & {\emptyset, {1}, {2}, {3}, {4}, {1, 2}, {1, 4}, {3, 2}, \\ {3, 4}, {1, 3}, {1, 2, 3}, {1, 2, 4}} \end{matrix}$

Let Ψ be a family of fuzzy sets defined on V = {1, 2, 3, 4} and ω ∈ [0, 1] ^V if and only if $ω_{[r]} satisfies {\begin{matrix} ω_{[r]} \in {\emptyset}, & 0.6 < r \leq 1, \\ ω_{[r]} \in I_{0.6}, & 0.4 < r \leq 0.6, \\ ω_{[r]} \in I_{0.4}, & 0.3 < r \leq 0.4, \\ ω_{[r]} \in I_{0.3}, & 0 < r \leq 0.3 . \end{matrix}$

It is easy to check that (V, Ψ) is a G-V fuzzy matroid. This approach in fact works in general case. It should be noted that this approach uses the representation theorem of fuzzy sets. In [28, 29], the representation theorem was generalized to intuitionistic fuzzy setting. We do not know whether this approach can be generalized to intuitionistic fuzzy matroids. In other words, to construct an intuitionistic fuzzy matroid from an given intuitionistic fuzzy graph will be our future work.

We now introduce the connection of approximations and fuzzy matroids. Let U be a nonempty finite set and E be an equivalence relation on U. The subset [x] _E = {y ∣ xEy} is the equivalence class containing x for every x ∈ U. Then E generates a partition U/R = {[x] _E ∣ x ∈ U}, namely, a family of pairwise disjoint subsets whose union is the universe. For any X ⊆ U, the (Pawlak) lower approximation and upper approximation of X, denoted by $\underline{apr} (X)$ and $\bar{apr} (X)$ respectively, are defined by ${\begin{matrix} \underline{apr} (X) = ⋃ {[x]_{E} ∣ x \in U, [x]_{E} \subseteq X}, \\ \bar{apr} (X) = ⋃ {[x]_{E} ∣ x \in U, [x]_{E} \cap X \neq \emptyset}, \end{matrix}$

The upper and lower approximations are fundamental concepts in rough sets. In [17], we pointed out that the upper and lower approximations coincide with the closure and inner of a partition matroid. Let $I = {X \subseteq U ∣ | X \cap Y | \leq 1 for every Y \in U / E} .$

Then $(E, I)$ is a matroid and called a partition matroid. Moreover, we have $\underline{apr} (X) = cl (X)$ and $\bar{apr} (X) = int (X)$ , where cl and int are closure and inner operators of the partition matroid respectively. In [7], Dubois and Prade defined fuzzy rough sets based on fuzzy partitions. Do the upper and lower approximations of a fuzzy set in the fuzzy rough set model equal the closure and inner of the fuzzy set in a G-V fuzzy matroid? Moreover, partitions have been generalized to intuitionistic fuzzy setting [24], our work in this paper will provide a new intuitionistic fuzzy rough set model with the closure and inner operators of an intuitionistic fuzzy matroid as its upper and lower approximation operators.

6 Conclusion

The purpose of this paper is to construct the basic concept of intuitionistic fuzzy matroid. Unlike existed work concerning fuzzy matroids (e.g., [1, 8], they all generalized the definition of matroid to intuitionistic fuzzy setting directly), we defined intuitionistic fuzzy matroids by an indirect approach. That is, the cuts of an intuitionistic fuzzy set system should possess the structure of crisp matroids. The connection of intuitionistic fuzzy matroids and G-V fuzzy matroids was given. Besides, the problems about construction of intuitionistic fuzzy matroids and connection with rough sets were presented. Finally, we present three interesting questions concerning intuitionistic fuzzy matroids, which are worthy of consideration in future.

Bases (i.e. maximal independent sets) are important for matroids. Many combinatorial optimization problems refer to bases of matroids. For example, the minimum spanning tree problem is exactly a special case of minimum-weight base problem for matroids. Thus, properties of bases for intuitionistic fuzzy matroids should be studied in detail.

As stated previously, we use an indirect approach to generalization of matroids in this paper. Do our intuitionistic fuzzy matroids have a corresponding version based on a direct approach?

Like topological spaces, matroids have several equivalent descriptions. Axioms for fuzzy matroids are also hot topics (see, e.g. [14 , 30]). Thus axiomatic descriptions of intuitionistic fuzzy matroids should be studied.

Footnotes

To be precise, X does not contain the edge set of a cycle, or equivalently, G [X], the subgraph induced by X, is a forest. Since the ground set of M (G) is the edge set of G, we shall refer to certain subgraphs of G when we mean just their edge sets. This commonplace practice should not cause confusion.

See Section 5 of [] for a comparison of five existed fuzzy matroids.

Note that τ satisfies (IFI1), thus Proposition 3.4 may help us when computing τ_<α,β>. For example, {a, b} ∈ τ_<0.5,0.5> holds obviously, by Proposition 3.4, we have τ_<0.5,0.5> = 2^{a,b}.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61772019), the Shaanxi Province Natural Science Foundation Research Project (No. 2017JM1036), the Fundamental Research Funds for the Central Universities (No. JB170702) and the China Postdoctoral Science Foundation (No. 2016M602851).

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