A generalization of G-V fuzzy matroids based on intuitionistic fuzzy sets

1 Introduction

The concept of matroid was firstly introduced by Whitney in 1935 [1]. Matroid theory is very important in discrete optimization and has been widely applied to combinatorial optimization problems such as shortest path, network flow problems etc. [2–4]. Since 1970, there has been a variety of extensions of matroid, which are concerned with fuzzy sets. For example,(L, M)-fuzzy matroid based on lattice-valued fuzzy set theory was introduced by Shi in [5]. For a (L, M)-fuzzy matroid, the family of the independent sets is the M-fuzzy family of independent L-fuzzy sets, L-matroids and M-fuzzifying matroids are its special forms [5–7]. Shi et al. also studied the bases axioms, circuits axioms and the rank functions of fuzzifying matroids or perfect [0, 1]-matroids [8–15]. Hsueh proposed one possible fuzzification of matroids which extends the independence axioms of matroids and preserves most basic properties of matroids [16]. Al-Hawary studied the properties of fuzzy regular-flats, fuzzy C-flats, fuzzy alternative-sets and furtherly introduced the notions of fuzzy C-matroids which is also a method for the fuzzifying of matroids [17, 18]. Another kind of fuzzy matroid based on fuzzy sets was introduced by Goetschel and Voxman (briefly, G - V fuzzy matroid) in [19]. They also studied the fuzzy bases, fuzzy circuits, fuzzy rank function, the greedy algorithm etc. [20–22]. Li, Zhang et al. presented the axioms for bases of both elementary G - V fuzzy matroids or closed regular G - V fuzzy matroids [23]. Li, Liu et al. studied connectedness of G - V fuzzy matroids and constructed the transitivity theorem concerning fuzzy circuits of G - V fuzzy matroids [24]. These are all the extensions of matroid based on fuzzy sets in recent years.

Intuitionistic fuzzy set was originally introduced by Atanassov [25], which is another important concept in this paper and extends the traditional fuzzy set. An intuitionistic fuzzy set is characterized by a membership function, a non-membership function and a hesitancy function. When the hesitancy function is equal to zero for each element, the intuitionistic fuzzy set reduces to a fuzzy set [25–27]. Xu and Yager proposed the intuitionistic fuzzy values which are composed of a membership degree, a non-membership degree and a hesitancy degree [28–30]. For ranking the intuitionistic fuzzy values, Hong and Choi proposed the accuracy function which emphasizes the amount of information that an intuitionistic fuzzy value contains [31]. Szmidt and Kacprzyk studied the representation of an intuitionistic fuzzy set and presented the distances between intuitionistic fuzzy sets [32]. Zhang and Xu presented the similarity function and a new method for ranking intuitionistic fuzzy values [33]. Li and Yi defined an intuitionistic fuzzy matroid by using an indirect approach and study their properties, which is the first time of the combination of matroids and intuitionistic fuzzy sets [34]. Motivated by these results, we have also been studying matroids and fuzzy matroids on intuitionistic fuzzy sets from the point of view relevant to G - V fuzzy matroid and find that some operators on intuitionistic fuzzy sets can’t be directly applied to matroids or traditional fuzzy matroids. Thus we defined the ranking, union and intersection between two intuitionistic fuzzy sets by using the similarity function h and the accuracy function H of the intuitionistic fuzzy value and proposed a different fuzzy matroid based on intuitionistic fuzzy sets and studied its induced matroid sequence and the rank function and their properties. This kind of fuzzy matroid is closely related to G - V fuzzy matroids and can be regarded as a generalization of G - V fuzzy matroids. Therefore, we call it G - V intuitionistic fuzzy matroid.

In this paper, some basic definitions and results are firstly introduced in Section 2 and then the concept of G - V intuitionistic fuzzy matroids is proposed in Section 3. In Section 4, an induced matroid sequence of G - V intuitionistic fuzzy matroids is investigated. Finally, an intuitionistic fuzzy rank function is proposed and its properties are studied in Section 5.

2 Preliminaries and notations

For the convenience of the readers, we recall some basic definitions and results concerning matroids theory, see [35–37].

Definition 2.1. Let E be a finite set and I be a nonempty family of subsets of E satisfying:

(I) If X ∈ I, and Y ⊂ X, then Y ∈ I.

(II) If X, Y ∈ I, and |Y| > |X|, then there is an x ∈ Y ∖ X such that X ∪ {x} ∈ I.

Then the pair M = (E, I) is a crisp matroid and the members of I are called the independent sets of M.

The rank function is very important in matroid theory. The concept and properties of the rank are as follows.

Definition 2.2. Let E is a fixed set and P (E) be the power set of E. Suppose that M = (E, I) is a (crisp) matroid. Then the mapping R : P (E) → {0, 1, 2, ⋯ , |E|} defined by R ( A ) = max { | B | | B ⊆ A and B ∈ I }(1) is called the rank function of M. R (A) is called the rank of A.

From the definition of the rank function above, we can easily obtain the following three properties of the rank function R, see [37].

Theorem 2.3. Suppose that R is the rank function of a crisp matroid M = (E, I) and A, B ∈ P (E). Then we have

If A ⊆ B, then R (A) ≤ R (B).

Submodularity is another most important property of the rank function R for a matroid M = (E, I). The definition is given below.

Definition 2.4. Let E be a finite set. A mapping σ : P (E) → [0, ∞) is called submodular if for any A, B ∈ P (E), σ ( A ) + σ ( B ) ≥ σ ( A ∩ B ) + σ ( A ∪ B )(2)

Theorem 2.5. The rank function R of a crisp matroid M = (E, I) is submodular.

Next, we introduce the concepts of fuzzy set and intuitionistic fuzzy set, see [25–33].

Definition 2.6. Let E is a fixed set. Then A = { ( x , μ A ( x ) ) | x ∈ E }(3) is called a fuzzy set, where μ _A  (x), 0 ≤ μ _A  (x) ≤1, is the membership degree of x to A. The family of all the fuzzy sets on E is denoted by FS (E).

Definition 2.7. Let E is a fixed set. Then A = { ( x , μ A ( x ) , ν A ( x ) ) | x ∈ E }(4) is called an intuitionistic fuzzy set, which assigns to each element x a membership degree μ _A  (x) and a non-membership degree ν _A  (x) with the conditions μ _A  (x) , ν _A  (x) ≥0 and 0 ≤ μ _A  (x) + ν _A  (x) ≤1 for each x ∈ E.

The family of all the intuitionistic fuzzy sets on E is denoted by IFS (E). Let π _A  (x) =1 - μ _A  (x) - ν _A  (x), which is called a hesitancy degree. Obviously, if π _A  (x) =0, then μ _A  (x) + ν _A  (x) =1. Thus an intuitionistic fuzzy set A reduces to a fuzzy set.

Xu and Yager called each triple (μ _α  (x) , ν _α  (x) , π _α  (x)), x ∈ E, an intuitionistic fuzzy value (or an intuitionistic fuzzy number). For convenience and the later research, an intuitionistic fuzzy set (μ _α , ν _α , π _α ) is abbreviated as (μ _α , π _α ) and the intuitionistic fuzzy value (μ _α  (x) , ν _α  (x) , π _α  (x)) is denoted by (μ _α  (x) , π _α  (x)) 1 .

To compare the size of two intuitionistic fuzzy sets, we introduce the accuracy function and the similarity function below, see [31–33].

Definition 2.8. Let (μ _α , π _α ) ∈ IFS (E) is an intuitionistic fuzzy set. Then for any x ∈ E, the accuracy function H of the intuitionistic fuzzy value (μ _α  (x) , π _α  (x)) is H ( μ α ( x ) , π α ( x ) ) = 1 - π α ( x ) .(5)

Definition 2.9. Let (μ _α , π _α ) ∈ IFS (E) be an intuitionistic fuzzy set. Then for any x ∈ E, the similarity function h of the intuitionistic fuzzy value (μ _α  (x) , π _α  (x)) is h ( μ α ( x ) , π α ( x ) ) = 1 - 1 - μ α ( x ) 1 - π α ( x ) .(6) In the special case π _α  (x) =0, we have h ( μ α ( x ) , π α ( x ) ) = μ α ( x ) .

Let E be a finite set and (μ _α , π _α ) , (μ _β , π _β ) ∈ IFS (E) be intuitionistic fuzzy sets. We now introduce the following notations and operators:

H_{(μ α ,π α )} (x) = H (μ _α  (x) , π _α  (x)), h_{(μ α ,π α )} (x) = h (μ _α  (x) , π _α  (x)), (μ _α , π _α ) (x) = (μ _α  (x) , π _α  (x)), x ∈ E.

Based on crisp matroid and intuitionistic fuzzy set, we propose the concept of G - V intuitionistic fuzzy matroid which is a generalization of G - V fuzzy matroid. Firstly, we define the union and intersection between two intuitionistic fuzzy sets.

Definition 3.1. Let (μ _α , π _α ) , (μ _β , π _β ) be two intuitionistic fuzzy sets and x ∈ E. (μ _γ , π _γ ) = (μ _α , π _α ) ∨ (μ _β , π _β ) and (μ _ω , π _ω ) = (μ _α , π _α ) ∧ (μ _β , π _β ) are called the union and intersection of (μ _α , π _α ) and (μ _β , π _β ), respectively, where (μ _γ , π _γ ) is defined by ( μ γ , π γ ) ( x ) = { ( μ α , π α ) ( x ) , if ( a 1 ) , ( μ α , π α ) ( x ) or ( μ β , π β ) ( x ) , if ( b 1 ) .(7) Where (a₁) h_{(μ α ,π α )} (x) ≥ h_{(μ β ,π β )} (x) and H_{(μ α ,π α )} (x) ≥ H_{(μ β ,π β )} (x) , (b₁) h_{(μ α ,π α )} (x) = h_{(μ β ,π β )} (x) and H_{(μ α ,π α )} (x) = H_{(μ β ,π β )} (x) .

and (μ _ω , π _ω ) is defined by ( μ ω , π ω ) ( x ) = { ( μ β , π β ) ( x ) , if ( a 2 ) , ( μ α , π α ) ( x ) or ( μ β , π β ) ( x ) , if ( b 2 ) .(8) Where (a₂) h_{(μ α ,π α )} (x) ≥ h_{(μ β ,π β )} (x) and H_{(μ α ,π α )} (x) ≥ H_{(μ β ,π β )} (x) , (b₂) h_{(μ α ,π α )} (x) = h_{(μ β ,π β )} (x) and H_{(μ α ,π α )} (x) = H_{(μ β ,π β )} (x) .

Now we combine matroids and intuitionistic fuzzy sets to propose the concept of G - V intuitionistic fuzzy matroids.

Definition 3.2. Let E be a finite set and let ψ ⊆ IFS (E) be a nonempty family of fuzzy sets.The pair IFM = (E, ψ) is called a G - V intuitionistic fuzzy matroid on E if it satisfies the following conditions:

(ψI) If (μ _α , π _α ) ∈ ψ, (μ _β , π _β ) ∈ IFS (E), and (μ _β , π _β ) ≺ (μ _α , π _α ), then (μ _β , π _β ) ∈ ψ.

(ψII) If (μ _α , π _α ) , (μ _β , π _β ) ∈ ψ, and |supp (μ _α , π _α ) | < |supp (μ _β , π _β ) |, then there exists (μ _ω , π _ω ) ∈ ψ, such that:

(a)(μ _α , π _α ) ≺ (μ _ω , π _ω ) ⪯ (μ _α , π _α ) ∨ (μ _β , π _β ),

(b) m (μ _ω , π _ω ) ≥ min {m (μ _α , π _α ) , m (μ _β , π _β )}.

For a G - V intuitionistic fuzzy matroid IFM = (E, ψ), the members of ψ are called independent intuitionistic fuzzy sets and ψ is called the family of independent intuitionistic fuzzy sets. Any member of IFS (E) but not belongs to ψ is called dependent intuitionistic fuzzy sets.

Obviously, if π _α  (x) =0 for any (μ _α , π _α ) ∈ IFS (E) and for any x ∈ E, then IFS (E) can be regarded as FS (E). Thus, G - V intuitionistic fuzzy matroids IFM = (E, ψ) reduces to G - V fuzzy matroids.

Note that the h-cardinality of intuitionistic fuzzy set (μ _α , π _α ) is ∣ ( μ α , π α ) ∣ = ∑ x ∈ E h ( μ α , π α ) ( x ) .(9)

Hence the independent sets of G - V intuitionistic fuzzy matroids are very different from those of G - V fuzzy matroids. The same is true for the dependent sets.

Example 3.3. Let V be the collection of some cities and E be the collection of the roads among the cities. Then G = (V, E) is a graph with the vertex set V and the edge set E. Suppose that n exports α _i  (i = 1, 2, ⋯ , n) evaluate the roads with satisfaction μ _{α i} , dissatisfaction ν _{α i} and uncertainty π _{α i} , where for each x ∈ E, μ _{α i}  (x) , π _{α i}  (x) ≥0 and μ _{α i}  (x) + ν _{α i}  (x) + π _{α i}  (x) =1. Then (μ _{α i} , π _{α i} ) is an intuitionistic fuzzy set on E. For any r, 0 < r ≤ 1, let E r = { e ∈ E | h ( μ α , π α ) ( e ) ≥ r } , L r = { F | F is a forest in the graph ( V , E r ) } , I r = { τ ( F ) | F ∈ L r } . where α represents an expert, h is the similarity function of the intuitionistic fuzzy value (μ _α  (e) , π _α  (e)) and τ (F) is the edge set of F.

From graph theory and Definition 2.1, it is easily to obtain that (E _r , I _r ) is a crisp matroid for any r, 0 < r ≤ 1. If ψ = { ( μ α , π α ) ∈ IFS ( E ) | C r ( μ α , π α ) ∈ I r , 0 < r ≤ 1 } .(10) Then, by Theorem 4.3 below, (E, ψ) is a G - V intuitionistic fuzzy matroid on E.

In this section, we first construct a family of a crisp matroid from a G - V intuitionistic fuzzy matroid.

Theorem 4.1. Suppose that IFM = (E, ψ) is a G - V intuitionistic fuzzy matroid. For each r, 0 ≤ r ≤ 1, let I r = { C r ( μ α , π α ) | ( μ α , π α ) ∈ ψ } .(11) Then for each r, 0 < r ≤ 1, M r = ( E , I r ) is a crisp matroid on E.

Proof. we should prove that I _r satisfies the conditions (I) and (II) of Definition 2.1.

(1) Suppose A ∈ I _r , B ⊆ A. Then there exists (μ _α , π _α ) ∈ ψ such that C _r  (μ _α , π _α ) = A. Let (μ _β , π _β ) ∈ IFS (E) and h ( μ β , π β ) ( x ) = { r , if x ∈ B , 0 , if x ∉ B .(12) Then C _r  (μ _β , π _β ) = B, and (μ _β , π _β ) ⪯ (μ _α , π _α ), it follows that (μ _β , π _β ) ∈ ψ, and hence B ∈ I _r . Then M _r  = (E, I _r ) satisfies condition (I) of Definition 2.1. (2) Suppose A, B ∈ I _r and |A| < |B|. Let (μ _α , π _α ) , (μ _β , π _β ) ∈ ψ, such that C _r  (μ _α , π _α ) = A and C _r  (μ _β , π _β ) = B. Let ( μ α ˆ , π α ˆ ) ∈ IFS ( E ) and h ( μ α ˆ , π α ˆ ) ( x ) = { r , if x ∈ A , 0 , if x ∉ A .(13) and let ( μ β ˆ , π β ˆ ) ∈ IFS ( E ) and h ( μ β ˆ , π β ˆ ) ( x ) = { r , if x ∈ B , 0 , if x ∉ B .(14) From the proof of (1), we have ( μ α ˆ , π α ˆ ) , ( μ β ˆ , π β ˆ ) ∈ ψ and

| supp ( μ α ˆ , π α ˆ ) | = | A | < | B | = | supp ( μ β ˆ , π β ˆ ) | .

By condition (ψII) of Definition 3.2, there exists (μ _ω , π _ω ) ∈ ψ such that

( μ α ˆ , π α ˆ ) ≺ ( μ ω , π ω ) ⪯ ( μ α ˆ , π α ˆ ) ∨ ( μ β ˆ , π β ˆ ) ,

m ( μ ω , π ω ) ≥ min { m ( μ α ˆ , π α ˆ ) , m ( μ β ˆ , π β ˆ ) } = r Obviously, h ( μ 1 , π 1 ) ( x ) = { r , if x ∈ A ∪ B , 0 , if x ∉ A ∪ B .(15) where x ∈ E and ( μ 1 , π 1 ) ( x ) = ( μ α ˆ , π α ˆ ) ( x ) ∨ ( μ β ˆ , π β ˆ ) ( x ) . Then there exists a set C satisfying A ⊂ C ⊆ A ∪ B such that h ( μ ω , π ω ) ( x ) = { r , if x ∈ C , 0 , if x ∉ C .(16) It implies that C ∈ I _r and I _r satisfies condition (II) of Definition 2.1.

Therefore, M _r  = (E, I _r ) is a crisp matroid.

Let E be a finite set, then the power set 2 ^E is a finite set. It implies that the number of family I _r of subsets of 2 ^E (where I _r is defined in Theorem 4.1) is finite. Then we have the following proposition.□

Proposition 4.2. Suppose that IFM = (E, ψ) is a G - V intuitionistic fuzzy matroid. For each r, 0 < r ≤ 1, let M _r  = (E, I _r ) is a crisp matroid on E defined in Theorem 4.1. Then there is a finite sequence r₀ < r₁ < ⋯ < r _n such that

(i) r₀ = 0, r _n  = 1.

(ii) I _s  ≠ {ϕ} if 0 < s ≤ r _n , I _s  = {ϕ} if s > r _n .

(iii) If r _i  < s, t < r_i+1, then I _s  = I _t , where 0 ≤ i ≤ n - 1.

(iv) If r _i  < s < r_i+1 < t < r_i+2, then I _s  ⊃ I _t , where 0 ≤ i ≤ n - 2.

Then the sequence r₀, r₁, r₂, ⋯ , r _n is called the fundamental sequence of IFM. Moreover, if we let r ¯ i = r i - 1 + r i 2 for any i, 1 ≤ i ≤ n, we can obtain a sequence of matroids M r ¯ n ⊂ M r ¯ n - 1 ⊂ ⋯ ⊂ M r 2 ¯ ⊂ M r 1 ¯ which is called the IFM - induced matroid sequence.

Note that for any 0 ≤ r ≤ 1,

C _r  (μ _α , π _α ) = {x ∈ E|h_{(μ α ,π α )} (x) ≥ r}

and I _r  = {C _r  (μ _α , π _α ) | (μ _α , π _α ) ∈ ψ}, then I _s  = {ϕ} not but I _s  = ϕ when s > r _n .

We construct a matroid sequence from a G - V intuitionistic fuzzy matroid above. On the contrary, we can also construct a G - V intuitionistic fuzzy matroid from a matroid sequence.

Theorem 4.3. Let E be a finite set, and let 0 = s₀ < s₁ < s₂ < ⋯ < s _n  ≤ 1 be a finite sequence. Suppose that M _{s 1} , M _{s 2} , ⋯ , M _{s n-1} , M _{s n} (M _{s i}  = (M, I _{s i} ) , 1 ≤ i ≤ n) is a matroid sequence on E such that I _{s i+1}  ⊂ I _{s i} , if 0 ≤ i ≤ n - 1. For any s, 0 < s ≤ 1, let I s = { I s i , if s i - 1 < s ≤ s i , 0 ≤ i ≤ n , { ϕ } , if s n < s ≤ 1 .(17) and let ψ ∗ = { ( μ α , π α ) ∈ IFS ( E ) | C s ( μ α , π α ) ∈ I s , 0 < s ≤ 1 } . Then IFM = (E, ψ^∗) is a G - V intuitionistic fuzzy matroid and its IFM - induced matroid sequence is M _{s n}  ⊂ M _{s n-1}  ⊂ ⋯ ⊂ M _{s 2}  ⊂ M _{s 1} , where for 1 ≤ i ≤ n, M _{s i}  = (M, I _{s i} ).

Proof. We show that condition (ψI) and (ψII) of Definition 3.2 hold.

(1) Suppose that (μ _α , π _α ) ∈ ψ^∗, (μ _β , π _β ) ∈ IFS (E) and (μ _β , π _β ) ⪯ (μ _α , π _α ). Then C _s  (μ _α , π _α ) ∈ I _s (0 < s ≤ 1), and C _s  (μ _β , π _β ) ⊆ C _s  (μ _α , π _α ) for each s. Then for each s, C _s  (μ _β , π _β ) ∈ I _s . It implies that (μ _β , π _β ) ∈ ψ^∗. Hence (E, ψ^∗) satisfies condition (ψI) of Definition 3.2.

(2) Suppose that (μ _α , π _α ) , (μ _β , π _β ) ∈ ψ^∗, and | supp ( μ α , π α ) | < | supp ( μ β , π β ) | . Then supp (μ _α , π _α ) ∈ I _p , supp (μ _β , π _β ) ∈ I _p ,where p = min {m (μ _α , π _α ) , m (μ _β , π _β )}. Because (E, I _p ) is a crisp matroid, there exists a set C ∈ I _p such that supp (μ _α , π _α ) ⊂ C ⊆ supp (μ _α , π _α ) ∪ supp (μ _β , π _β ). Let h ( μ ω , π ω ) ( x ) = { h ( μ α , π α ) ( x ) , if ( a 3 ) , p , if ( b 3 ) , 0 , otherwise .(18) Where

(a₃) x ∈ supp (μ _α , π _α ) ,

(b₃) x ∈ C ∖ supp (μ _α , π _α ) . Obviously,

m ( μ ω , π ω ) = p = min { m ( μ α , π α ) , m ( μ β , π β ) } . It implies that (μ _ω , π _ω ) satisfies condition (ψII) of Definition 3.2.□

Therefore, (E, ψ^∗) is a G - V intuitionistic fuzzy matroid and obviously M _{s n}  ⊂ M _{s n-1}  ⊂ ⋯ ⊂ M _{s 2}  ⊂ M _{s 1} is an IFM - induced matroid sequence.

Next, we construct a family of independent intuitionistic fuzzy sets, which is equal to the family of independent intuitionistic fuzzy sets of a G - V intuitionistic fuzzy matroid.

Theorem 4.4. Let IFM = (E, ψ) be a G - V intuitionistic fuzzy matroid. Suppose that for each r, 0 < r ≤ 1, M _r  = (E, I _r ) is a matroid on E defined by Theorem 4.1. Let ψ ∗ = { ( μ α , π α ) ∈ IFS ( E ) | C r ( μ α , π α ) ∈ I r , 0 < r ≤ 1 } .(19) Then ψ = ψ^∗.

Proof. (1) Let (μ _α , π _α ) ∈ ψ. Then we have that C _r  (μ _α , π _α ) ∈ I _r for each r, 0 < r ≤ 1. It follows that (μ _α , π _α ) ∈ ψ^∗. Then ψ ⊆ ψ^∗.

(2) Suppose that (μ _α , π _α ) ∈ ψ^∗ and let R⁺ (μ _α , π _α ) = {t₁, t₂, ⋯ , t _q } is the nonzero h - range of (μ _α , π _α ) with t₁ > t₂ > ⋯ > t _q . Obviously, C _{t i}  (μ _α , π _α ) ∈ I _{t i} for 1 ≤ i ≤ q and C _{t i}  (μ _α , π _α ) ⊆ C _{t i+1}  (μ _α , π _α ) for 1 ≤ i ≤ q - 1. Then for 1 ≤ i ≤ q, we define (μ _{β i} , π _{β i} ) ∈ IFS (E) by h ( μ β i , π β i ) ( x ) = { t i , if x ∈ C t i ( μ α , π α ) , 0 , otherwise .(20) Since that C _{t i}  (μ _α , π _α ) ∈ I _{t i} for 1 ≤ i ≤ q, it follows that for 1 ≤ i ≤ q, we have (μ _{β i} , π _{β i} ) ∈ ψ and (μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β q} , π _{β q} ) = (μ _α , π _α ).

Let supp ( μ β 1 , π β 1 ) = { e 1 , e 2 , ⋯ , e n 1 } , supp ( μ β 2 , π β 2 ) = { e 1 , e 2 , ⋯ , e n 1 , e n 1 + 1 , ⋯ , e n 2 } , ⋯ , ⋯ , supp ( μ β q , π β q ) = { e 1 , e 2 , ⋯ , e n 1 , e n 1 + 1 , ⋯ , e n 2 , e n 2 + 1 , ⋯ , e n q } = supp ( μ α , π α ) .

Then n₁ < n₂ < ⋯ < n _q .

We proceed by induction on q to show that (μ _α , π _α ) = (μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β q} , π _{β q} ) ∈ ψ

Obviously, for q = 1, (μ _{β 1} , π _{β 1} ) ∈ ψ. We will show that (μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β k} , π _{β k} ) ∈ ψ. if (μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β k-1} , π _{β k-1} ) ∈ ψ, where k - 1 < q.

Let ( μ ω 1 ( k ) , π ω 1 ( k ) ) ∈ IFS ( E ) and h ( μ ω 1 ( k ) , π ω 1 ( k ) ) ( x ) = { t k , if ( a 4 ) , 0 , if ( b 4 ) .(21) where (a₄) x ∈ {e₁, e₂, ⋯ , e _{n k-1} , e_nk-1+1} , (b₄) otherwise.

Obviously, ( μ ω 1 ( k ) , π ω 1 ( k ) ) ≤ ( μ β k , π β k ) . Then ( μ ω 1 ( k ) , π ω 1 ( k ) ) ∈ ψ . Let h ( μ δ 1 , π δ 1 ) ( x ) = { h ( μ α , π α ) ( x ) , if x = e n k - 1 + 1 0 , otherwise .(22) where (μ _{δ 1} , π _{δ 1} ) ∈ IFS (E). Let ( μ z 1 ( k ) , π z 1 ( k ) ) = ( μ β 1 , π β 1 ) ∨ ( μ β 2 , π β 2 ) ∨ ⋯ ∨ ( μ β k - 1 , π β k - 1 ) ∨ ( μ δ 1 , π δ 1 ) .(23) By the induction assumption, (μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β k-1} , π _{β k-1} ) ∈ ψ. Since supp ( ( μ β 1 , π β 1 ) ∨ ( μ β 2 , π β 2 ) ∨ ⋯ ∨ ( μ β k - 1 , π β k - 1 ) ) = { e 1 , e 2 , ⋯ , e n k - 1 } and m ((μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β k-1} , π _{β k-1} )) > t _k . Then for the intuitionistic fuzzy sets (μ _{β 1} , π _{β 1} ) ∨ (μ _{β 2} , π _{β 2} ) ∨ ⋯ ∨ (μ _{β k-1} , π _{β k-1} ) and ( μ ω 1 ( k ) , π ω 1 ( k ) ) , we can get that ( μ z 1 ( k ) , π z 1 ( k ) ) ∈ ψ from condition (ψII) of Definition 3.2.

If n_k-1 + 1 = n _k , then ( μ β 1 , π β 1 ) ∨ ⋯ ∨ ( μ β k - 1 , π β k - 1 ) ∨ ( μ β k , π β k ) = ( μ β 1 , π β 1 ) ∨ ⋯ ∨ ( μ β k - 1 , π β k - 1 ) ∨ ( μ δ 1 , π δ 1 ) = ( μ z 1 ( k ) , π z 1 ( k ) ) ∈ ψ It follows that (μ _α , π _α ) ∈ ψ. It implies that ψ^∗ ⊆ ψ.

If n_k-1 + 1 < n _k , then let ( μ ω 2 ( k ) , π ω 2 ( k ) ) ∈ IFS ( E ) and h ( μ ω 2 ( k ) , π ω 2 ( k ) ) ( x ) = { t k , if ( a 5 ) 0 , if ( b 5 )(24) Where (a₅)x ∈ {e₁, e₂, ⋯ , e_nk-1+1, e_nk-1+2}, (b₅) otherwise.

Obviously, ( μ ω 2 ( k ) , π ω 2 ( k ) ⪯ ( μ β k , π β k ) . Then ( μ ω 2 ( k ) , π ω 2 ( k ) ) ∈ ψ

Let h ( μ δ 2 , π δ 2 ) ( x ) = { h ( μ α , π α ) ( x ) = t k , if ( a 6 ) 0 , if ( b 6 )(25) where (a₆) x = e_nk-1+2, (b₆) otherwise, and (μ _{δ 2} , π _{δ 2} ) ∈ IFS (E).

Let ( μ z 2 ( k ) , π z 2 ( k ) ) = ( μ z 1 ( k ) , π z 1 ( k ) ) ∨ ( μ δ 2 , π δ 2 ) , because

supp ( μ z 1 ( k ) , π z 1 ( k ) ) = { e 1 , e 2 , ⋯ , e n k - 1 , e n k - 1 + 1 } and m ( μ z 1 ( k ) , π z 1 ( k ) ) = t k , ( μ z 2 ( k ) , π z 2 ( k ) ) ∈ ψ can be attained from condition (ψII) of Definition 3.2 applied to the two intuitionistic fuzzy sets ( μ z 1 ( k ) , π z 1 ( k ) ) and ( μ ω 2 ( k ) , π ω 2 ( k ) ) .

If n_k-1 + 2 = n _k , then ( μ β 1 , π β 1 ) ∨ ⋯ ∨ ( μ β k - 1 , π β k - 1 ) ∨ ( μ β k , π β k ) = ( μ z 1 ( k ) , π z 1 ( k ) ) ∨ ( μ δ 2 , π δ 2 ) = ( μ z 2 ( k ) , π z 2 ( k ) ) ∈ ψ It follows that (μ _α , π _α ) ∈ ψ. It implies that ψ^∗ ⊆ ψ.

If n_k-1 + 2 < n _k , then we can proceed as before and eventually get an Intuitionistic fuzzy set ( μ z m ( k ) , π z m ( k ) ) ∈ ψ such that ( μ z m ( k ) , π z m ( k ) ) = ( μ β 1 , π β 1 ) ∨ ( μ β 2 , π β 2 ) ∨ ⋯ ∨ ( μ β k , π β k ) .(26) It follows that (μ _α , π _α ) ∈ ψ. It implies that ψ^∗ ⊆ ψ. Therefore, the proof of the theorem is finished.

In this section, the concept of rank function for a G - V intuitionistic fuzzy matroid is proposed, which is an extension of the rank function for a G - V fuzzy matroid.

Definition 5.1. Suppose that IFM = (E, ψ) is a G - V intuitionistic fuzzy matroid. Let (μ _α , π _α ) ∈ IFS (E). The rank function ρ of a G - V intuitionistic fuzzy matroid is a mapping ρ :  IFS (E) ↦ [0, + ∞) defined by ρ ( μ α , π α ) = sup { | ( μ β , π β ) | | ( μ β , π β ) ⪯ ( μ α , π α ) , ( μ β , π β ) ∈ ψ } ,(27) where ∣ (μ _β , π _β ) ∣ = ∑_x∈Eh_{(μ β ,π β )} (x).

It is easy to check that ρ satisfies the following properties which are similar to the properties of Theorem 2.3.

(i) If (μ _α , π _α ) , (μ _β , π _β ) ∈ IFS (E), and (μ _α , π _α ) ⪯ (μ _β , π _β ), then ρ (μ _α , π _α ) ≤ ρ (μ _β , π _β )

(ii) ρ (μ _α , π _α ) ≤ | (μ _α , π _α ) | for any (μ _α , π _α ) ∈ IFS (E)

(iii) If (μ _α , π _α ) ∈ ψ, then ρ (μ _α , π _α ) = | (μ _α , π _α ) |.

The method of defining the closure of G - V fuzzy matroids is ingenious, which is applied to define the concept of the closure of G - V intuitionistic fuzzy matroids below.

Definition 5.2. Let IFM = (E, ψ) be a G - V intuitionistic fuzzy matroid with the fundamental sequence 0 = r₀ < r₁ < r₂ < ⋯ < r _n  ≤ 1. I _r (0 < r ≤ 1) is defined as in Theorem 4.1, and let I ¯ r = { I r ¯ i , if r i - 1 < r ≤ r i I r , if r > r n(28) Where r ¯ i = r i - 1 + r i 2 . Let ψ ¯ = { ( μ α , π α ) ∈ IFS ( E ) | C r ( μ α , π α ) ∈ I ¯ r , 0 < r ≤ 1 } .(29) Then IFM ¯ = ( E , ψ ¯ ) is called the closure of IFM.

It is clearly that if r_i-1 < r < r _i , then I ¯ r = I r .

From Theorem 4.3, the following conclusion can be derived directly.

Theorem 5.3. Suppose that IFM = (E, ψ) is a G - V intuitionistic fuzzy matroid. Then the closure IFM ¯ = ( E , ψ ¯ ) of IFM is a G - V intuitionistic fuzzy matroid.

From theorem 4.3, we can also obtain that IFM and IFM ¯ have the same fundamental sequence.

Definition 5.4. Suppose that IFM = (E, ψ) is a G - V intuitionistic fuzzy matroid with the fundamental sequence 0 = r₀ < r₁ < r₂ < ⋯ < r _n  ≤ 1. If I _r  = I _{r i} for any r_i-1 < r ≤ r _i (0 ≤ i ≤ n), then IFM = (E, ψ) is called a closed G - V intuitionistic fuzzy matroid.

Obviously, the closure of a G - V intuitionistic fuzzy matroid IFM is the smallest closed G - V intuitionistic fuzzy matroid which contains IFM.

Proposition 5.5. Let IFM ¯ = ( E , ψ ¯ ) be the closure of a G - V intuitionistic fuzzy matroid IFM = (E, ψ), ( μ α , π α ) ∈ ψ ¯ . ρ ¯ and ρ are the rank functions of IFM ¯ and IFM, respectively. Then ρ ¯ = ρ .

Proof. (1) For any (μ _α , π _α ) ∈ ψ, we have C r ( μ α , π α ) ∈ I ¯ r for each 0 < r ≤ 1. Then ( μ α , π α ) ∈ ψ ¯ . It follows that ψ ⊆ ψ ¯ . This implies that ρ ( μ α , π α ) ≤ ρ ¯ ( μ α , π α ) .

(2) We will show that ρ ¯ ( μ α , π α ) ≤ ρ ( μ α , π α ) for each (μ _α , π _α ) ∈ IFS (E).

Suppose that ( μ β , π β ) ∈ ψ ¯ and (μ _β , π _β ) ⪯ (μ _α , π _α ). Let R + ( μ β , π β ) = { h 1 , h 2 , ⋯ , h k } , where ( μ β , π β ) ∈ ψ ¯ and h₁ < h₂ < ⋯ < h _k .

Let ɛ > 0 and η = min { ɛ / | E | , min 1 ≤ i ≤ k - 1 { h i + 1 - h i 2 } } .

Let 0 = r₀ < r₁ < r₂ < ⋯ < r _n  ≤ 1 be the fundamental sequence of IFM and for any x ∈ E, let (μ_β′, π_β′) ∈ IFS (E) and h ( μ β ′ , π β ′ ) = { h ( μ β , π β ) ( x ) , if ( a 7 ) h ( μ β , π β ) ( x ) - η , if ( b 7 )(30) where

(a₇) h_{(μ β ,π β )} (x) ∉ {r₁, r₂, ⋯ , r _n },

(b₇) h_{(μ β ,π β )} (x) ∈ {r₁, r₂, ⋯ , r _n }. Then ( μ β ′ , π β ′ ) ∈ ψ ¯ , (μ_β′, π_β′) ≺ (μ _α , π _α ) and | ( μ β ′ , π β ′ ) | ≤ | ( μ β , π β ) | = ∑ h ( μ β , π β ) ∈ { r 1 , r 2 , ⋯ , r n } h ( μ β , π β ) + ∑ h ( μ β , π β ) ∉ { r 1 , r 2 , ⋯ , r n } h ( μ β , π β ) = ∑ h ( μ β ′ , π β ′ ) ∈ { r 1 , r 2 , ⋯ , r n } h ( μ β ′ , π β ′ ) + ∑ h ( μ β ′ , π β ′ ) ∉ { r 1 , r 2 , ⋯ , r n } ( h ( μ β ′ , π β ′ ) + η ) ≤ | ( μ β ′ , π β ′ ) | + ɛ

Let {(μ _{β i} , π _{β i} )} be a sequence of independent intuitionistic fuzzy sets in ψ ¯ such that for any i, we have (μ _{β i} , π _{β i} ) ⪯ (μ _α , π _α ) and lim i → ∞ | ( μ β i , π β i ) | = ρ ¯ ( μ α , π α ) Then for each (μ _{β i} , π _{β i} ), there is an independent intuitionistic fuzzy set (μ_{β′ i} , π_{β′ i} ) in ψ such that (μ_{β′ i} , π_{β′ i} ) ⪯ (μ _α , π _α ) and | ( μ β i , π β i ) | ≤ | ( μ β ′ i , π β ′ i ) | + ɛ Then we have ρ ¯ ( μ α , π α ) < ( sup { ( μ β ′ i , π β ′ i ) } ) + ɛ ≤ ρ ( μ α , π α ) + ɛ . Since ɛ > 0 was arbitrarily chosen, then ρ ¯ ( μ α , π α ) ≤ ρ ( μ α , π α ) .

Therefore, the proof is completed.

We now show that the intuitionistic fuzzy rank function ρ of a closed G - V intuitionistic fuzzy matroid IFM = (E, ψ) is submodular as follows.

Suppose that IFM = (E, ψ) is a closed G - V intuitionistic fuzzy matroid with intuitionistic fuzzy rank function ρ and fundamental sequence 0 = r₀ < r₁ < r₂ < ⋯ < r _n  ≤ 1. Let function ρ ˆ : IFS ( E ) → [ 0 , ∞ ) be submodular. ρ ˆ is defined as below.

Suppose further that (μ _α , π _α ) ∈ IFS (E) with R⁺ (μ _α , π _α ) = {h₁, h₂, ⋯ , h _s }. Let { a 1 , a 2 , ⋯ , a m } = { r 1 , r 2 , ⋯ , r n } ∪ { h 1 , h 2 , ⋯ , h s } ,(31) where a _{j i}  = r _i , 0 = a₀ < a₁ < a₂ < ⋯ < a _m . And let R _i be the rank function of a crisp matroid M = (E, I _{r i} ).

It is obvious that for each positive integer j, 1 ≤ j ≤ j _n  = r _n , there is a corresponding integer i, 1 ≤ i ≤ n, such that r_i-1=ai-1 ≤ a_j-1 < a _j  ≤ a _{j i}  = r _i . Then the pair (i, j) here is called a correspondence pair. Let q j ( μ α , π α ) = { ( a j - a j - 1 ) R i ( C a j ( μ α , π α ) ) , if ( a 8 ) 0 , if ( b 8 ) .(32) where

(a₈) a _j  ≤ r _n  and  (i, j)  is a correspondence pair,

(b₈) a _j  > r _n .

For a closed G - V intuitionistic fuzzy matroid IFM = (E, ψ), if a_j-1 < η ≤ a _j for every j, 1 ≤ j ≤ m, and for each η, then we have C _η  (μ _α , π _α ) = C _{a j}  (μ _α , π _α ).

Now function ρ ˆ : IFS ( E ) → [ 0 , ∞ ) is defined by ρ ˆ ( μ α , π α ) = ∑ j = 1 m q j ( μ α , π α )(33) Thus, from the definition of q _j  (μ _α , π _α ), the following proposition can be obtained.□

Proposition 5.6. Suppose that IFM = (E, ψ) is a closed G - V intuitionistic fuzzy matroid with fundamental sequence 0 = r₀ < r₁ < r₂ < ⋯ < r _n  ≤ 1, and {a₁, a₂, ⋯ , a _m } ⊆ {d₁, d₂, ⋯ , d _t }, where 0 < d₁ < d₂ < ⋯ < d _t  ≤ 1 and a₁ < a₂ < ⋯ < a _m is defined as above. Let d _{k i}  = r _i for 1 ≤ i ≤ n and let (i, k) be a correspondence pair if d _{k i-1}  ≤ d_k-1 < d _k  ≤ d _{k i} .

For each k, 1 ≤ k ≤ t. If q k ∗ : IFS ( E ) → R 1 is defined by q k ∗ ( μ α , π α ) = { ( d k - d k - 1 ) R i ( C d k ( μ α , π α ) ) , if ( a 9 ) , 0 , if ( b 9 ) .(34) Where (a₉) d _k  ≤ r _n and (i, k) is a correspondence pair, (b₉) d _k  > r _n .

Then for each (μ _α , π _α ) ∈ IFS (E) ∑ j = 1 m q j ( μ α , π α ) = ∑ k = 1 t q k ∗ ( μ α , π α ) ,(35)

The intuitionistic fuzzy rank function above satisfies submodular.

Theorem 5.7. Suppose that IFM = (E, ψ) is a closed G - V intuitionistic fuzzy matroid with fundamental sequence 0 = r₀ < r₁ < r₂ < ⋯ < r _n  ≤ 1. If ρ ˆ : IFS ( E ) ↦ [ 0 , + ∞ ) is defined by (33), then ρ ˆ is submodular, i.e. for any (μ _α , π _α ) , (μ _β , π _β ) ∈ IFS (E), there is