Bases of G-V Intuitionistic Fuzzy Matroids

Abstract

The purpose of this paper is to study intuitionistic fuzzy bases ($IFBs$) and the intuitive structure of a $G-V$$IFM$. Firstly, the intuitionistic fuzzy basis ($IFB$) of a $G-V$$IFM$ is defined; then the h-range and properties of an $IFB$ are presented and a necessary and sufficient condition of a closed $G-V$$IFM$ is studied. Especially, a necessary and sufficient condition of judging an $IFB$ is presented and the intuitive tree structure of a closed $G-V$$IFM$ is proposed and its properties are discussed.

1. Introduction

Whitney’s 1935 article laid the groundwork for the field of combinatorial geometries and matroid [1]. Matroid theory has been widely applied to combinatorial mathematics, combinatorial optimization and group theory [2,3,4,5,6,7,8]. Based on the fuzzy set theory proposed by Zadeh in 1965 [9], matroid theory has been generalized to various forms related to fuzzy sets. Shi [10,11] proposed the $(L,M)$-fuzzy matroid based on latticevalued fuzzy set theory and studied the base axioms of fuzzitying matroids [12,13,14]. Hsueh presented a fuzzification of matroids which extends the independence axioms of matroids [15]. Al-Hawary introduced a method to the fuzzifying of matroids which is called fuzzy C-matroids [16,17]. In 1988, Goetschel and Voxman proposed an important fuzzy matroid (briefly, $G-V$ fuzzy matroid) in [18]. They further studied some important concepts and their properties, such as the fuzzy bases and the fuzzy rank function [19,20,21,22]. Following them, some scholars studied the axioms, the connectedness and the structure of $G-V$ fuzzy matroid, etc. [23,24,25].

The intuitionistic fuzzy set ($IFS$), introduced by Atanassov originally in 1983 [26] and made widely accessible in 1986 [27], is a generalization of Zadeh’s fuzzy set. An $IFS$ of each element is an ordered pair which is called an intuitionistic fuzzy value ($IFV$) and each $IFV$ is characterized by a membership degree, a nonmembership degree and a hesitancy degree. From then on, many scholars were attracted to study the $IFS$ and obtained a lot of valuable results. For ranking the $IFSs$, Hong and Choi proposed the accuracy function in 2000 [28] and Szmidt and Kacprzyk proposed a similarity function of $IFSs$ in 2004 [29]. Based on the accuracy function and the similarity function, Zhang and Xu introduced a new method for ranking $IFSs$ in 2012 [30]. In 2013, Rangasamy et al. proposed a method by ranking to be done using the scores and accuracy for finding the shortest hyperpath in an intuitionistic fuzzy weighted hypergraph [31]. Some other scholars studied the aggregation operators and fuzzy clustering of $IFSs$ [32,33,34]. After decades of effort from scholars, the relevant achievements of intuitionistic fuzzy theory became very rich. In 1999, Atanassov completed his first monograph which discussed the concept and operators of $IFSs$, the interval valued $IFSs$, some other extensions of $IFSs$, the elements of $IFSs$ and the applications of $IFSs$[35]. There are some other scholars’ results worthy of learning and researching; see [36,37,38]. In 2017, Li and Yi proposed an intuitionistic fuzzy matroid based on matroids and intuitionistic fuzzy sets [39]. In [40], Li et al. extended $G-V$ fuzzy matroids and introduced a $G-V$ intuitionistic fuzzy matroid and studied the induced matroid sequence and the rank function. In this paper, based on the literature [19,25,40], we study the bases and the structure of a $G-V$ intuitionistic fuzzy matroid (briefly, $G-V$$IFM$), which are actually generalizations of some conclusions of $G-V$ fuzzy matroid.

This paper is arranged as follows. Some basic definitions and results are introduced in Section 2. The $IFBs$ of a $G-V$$IFM$ are studied in Section 3. The judgment of an $IFB$ is investigated in Section 4. Finally, we propose the tree structure of a closed $G-V$$IFM$ and study its properties in Section 5.

2. Preliminaries and Notations

We introduce some basic and useful concepts related to matroid theory here; see [41,42]. Firstly, we introduce the concept of the matroid.

Definition 1.

Let I be a nonempty family of subsets of a finite set E and satisfy:

1.

$\varnothing \in I$.

2.

If$X\in I$, and$Y\subset X$, then$Y\in I$.

3.

If$X,Y\in I$, and$\left|Y\right|>\left|X\right|$, then there exists an$x\in Y\backslash X$such that$X\cup \left\{x\right\}\in I$.

Then the pair$M=(E,I)$is called a matroid (or a crisp matroid). For any$A\subseteq E$, if$A\in I$, then A is called an independent set; otherwise A is called a dependent set.

In matroid theory, rank function and its submodularity are very important. They are defined as follows.

Definition 2.

Let$P\left(E\right)$be the power set of finite set E and$M=(E,I)$be a matroid. R is called rank function of M, where$R:P\left(E\right)\to \{0,1,2,\cdots ,|E\left|\right\}$is a mapping and is defined as follows:

$$R\left(X\right)=max\left\{\right|Y\left|\right|Y\subseteq X,andY\in I\}.$$

From the definition of R, the following properties can be easily obtained.

• If $X\subseteq Y$, then $R\left(X\right)\le R\left(Y\right)$;
• $R\left(X\right)\le \left|X\right|$ for any $X\in P\left(E\right)$;
• If $X\in I$, then $R\left(X\right)=\left|X\right|$,

where $X,Y\in P\left(E\right).$

Definition 3.

Let $\sigma :P\left(E\right)\to [0,\infty )$ be a mapping, where $P\left(E\right)$ is the power set of finite set E. σ is called submodular if

$$\sigma \left(X\right)+\sigma \left(Y\right)\ge \sigma (X\cap Y)+\sigma (X\cup Y),$$

for each $X,Y\in P\left(E\right)$.

Theorem 1.

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

Next, some concepts and notations concerning fuzzy sets or intuitionistic fuzzy sets are cited; see [9,18,19,26,27,28,29,30,31,32,33,34,35,36,37,38,40].

Definition 4.

Let X be a fixed set. Then

$$A=\left\{(x,{\mu }_A\left(x\right))\right|x\in X\}$$

is called a fuzzy set, where ${\mu }_A\left(x\right)$ is the membership degree of x to A, $0\le {\mu }_A\left(x\right)\le 1$. The collection of fuzzy sets on X is denoted by $FS\left(X\right)$.

Definition 5.

Let X be a fixed set. Then

$$A=\left\{(x,{\mu }_A\left(x\right),{\nu }_A\left(x\right))\right|x\in X\}$$

is called an $IFS$ (i.e., intuitionistic fuzzy set). For any $x\in X$, ${\mu }_A\left(x\right)$, ${\nu }_A\left(x\right)$ and ${\pi }_A\left(x\right)$ are called membership degree, non-membership degree and hesitancy degree, respectively, where ${\mu }_A\left(x\right),{\nu }_A\left(x\right),{\pi }_A\left(x\right)\ge 0$ and ${\mu }_A\left(x\right)+{\nu }_A\left(x\right)+{\pi }_A\left(x\right)=1$. The collection of $IFSs$ on X is denoted by $IFS\left(X\right)$. If for all $x\in X$, ${\pi }_A\left(x\right)=0$, then ${\mu }_A\left(x\right)+{\nu }_A\left(x\right)=1$ and $IFS$A is reduced to a fuzzy set. In this paper, we use $({\mu }_{\alpha },{\nu }_{\alpha },{\pi }_{\alpha })$ to denote intuitionistic fuzzy set and $({\mu }_{\alpha }\left(x\right),{\nu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$ to denote intuitionistic fuzzy value.

For convenience and suitable for the study of $G-V$ intuitionistic fuzzy matroids later, an $IFS$$({\mu }_{\alpha },{\nu }_{\alpha },{\pi }_{\alpha })$ is abbreviated as $({\mu }_{\alpha },{\pi }_{\alpha })$ and an $IFV$$({\mu }_{\alpha }\left(x\right),{\nu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$ is denoted by $({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$. Note that this notation is different from that in Definition 5.

Definition 6.

Let $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(X\right)$ be an $IFS$. Then the accuracy function H of $({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right)),(x\in X)$ is denoted by

$$H({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=1-{\pi }_{\alpha }\left(x\right)$$

Definition 7.

Let $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(X\right)$ be an $IFS$. Then the similarity function h of $({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$ for any $x\in X$ is

$$h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=1-\frac{1-{\mu }_{\alpha }\left(x\right)}{1+{\pi }_{\alpha }\left(x\right).}$$

In the special case ${\pi }_{\alpha }\left(x\right)=0$, we have $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))={\mu }_{\alpha }\left(x\right)$.

Let X be a finite set and $({\mu }_{\alpha },{\pi }_{\alpha }),({\mu }_{\beta },{\pi }_{\beta })\in IFS\left(X\right)$ be $IFSs$ and $x\in X$. We now introduce the following notation and results; see [40]:

• $H_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)=H({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$.
  $h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)=h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$.
  $({\mu }_{\alpha },{\pi }_{\alpha })\left(x\right)=({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$.
• $({\mu }_{\alpha },0)=({\mu }_{\alpha },{\pi }_{\alpha })$ if ${\pi }_{\alpha }\left(x\right)=0$ for any $x\in X$.
• $h({\mu }_{\alpha },{\pi }_{\alpha })\le h({\mu }_{\beta },{\pi }_{\beta })$: for any $x\in X$, $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))\le h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$.
  $h({\mu }_{\alpha },{\pi }_{\alpha })=h({\mu }_{\beta },{\pi }_{\beta })$: for any $x\in X$, $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$.
  $h({\mu }_{\alpha },{\pi }_{\alpha })<h({\mu }_{\beta },{\pi }_{\beta })$: $h({\mu }_{\alpha },{\pi }_{\alpha })\le h({\mu }_{\beta },{\pi }_{\beta })$ and $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))<h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$ for some $x\in X.$
• $H({\mu }_{\alpha },{\pi }_{\alpha })\le H({\mu }_{\beta },{\pi }_{\beta })$: for any $x\in X$, $H({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))\le H({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$.
  $H({\mu }_{\alpha },{\pi }_{\alpha })=H({\mu }_{\beta },{\pi }_{\beta })$: for any $x\in X$, $H({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=H({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$.
  $H({\mu }_{\alpha },{\pi }_{\alpha })<H({\mu }_{\beta },{\pi }_{\beta })$:$H({\mu }_{\alpha },{\pi }_{\alpha })\le H({\mu }_{\beta },{\pi }_{\beta })$ and $H({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))<H({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$ for some $x\in X$.
• $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })$:$h({\mu }_{\alpha },{\pi }_{\alpha })\le h({\mu }_{\beta },{\pi }_{\beta })$ and $H({\mu }_{\alpha },{\pi }_{\alpha })\le H({\mu }_{\beta },{\pi }_{\beta })$.
  $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\beta },{\pi }_{\beta })$:$h({\mu }_{\alpha },{\pi }_{\alpha })<h({\mu }_{\beta },{\pi }_{\beta })$ and $H({\mu }_{\alpha },{\pi }_{\alpha })\le H({\mu }_{\beta },{\pi }_{\beta })$.
  $({\mu }_{\alpha },{\pi }_{\alpha })=({\mu }_{\beta },{\pi }_{\beta })$:$h({\mu }_{\alpha },{\pi }_{\alpha })=h({\mu }_{\beta },{\pi }_{\beta })$ and $H({\mu }_{\alpha },{\pi }_{\alpha })=H({\mu }_{\beta },{\pi }_{\beta })$.
• supp$({\mu }_{\alpha },{\pi }_{\alpha })=\{x\in X|h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))>0\}$.
• $m({\mu }_{\alpha },{\pi }_{\alpha })=inf\left\{h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))\right|x\in$supp$({\mu }_{\alpha },{\pi }_{\alpha })\}$.
• $C_r({\mu }_{\alpha },{\pi }_{\alpha })=\{x\in X|h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))\ge r\}$, where $0\le r\le 1$.
• $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })=\left\{h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))\right|h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))>0,x\in X\}$ is called the positive $h-range$ of $({\mu }_{\alpha },{\pi }_{\alpha })$.
• $|({\mu }_{\alpha },{\pi }_{\alpha })|={\sum }_{x\in X}h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$ is called the "cardinality" of an $IFS$.

Definition 8.

Let $({\mu }_{\alpha },{\pi }_{\alpha }),({\mu }_{\beta },{\pi }_{\beta })$ be two intuitionistic fuzzy sets, $x\in X$. $({\mu }_{\gamma },{\pi }_{\gamma })=({\mu }_{\alpha },{\pi }_{\alpha })\vee ({\mu }_{\beta },{\pi }_{\beta })$ and $({\mu }_{\omega },{\pi }_{\omega })=({\mu }_{\alpha },{\pi }_{\alpha })\wedge ({\mu }_{\beta },{\pi }_{\beta })$ are called the union and intersection of $({\mu }_{\alpha },{\pi }_{\alpha })$ and $({\mu }_{\beta },{\pi }_{\beta })$, respectively, where $({\mu }_{\gamma },{\pi }_{\gamma })$ is defined by

$$({\mu }_{\gamma },{\pi }_{\gamma })\left(x\right)=\left\{\begin{array}{ccc}\hfill ({\mu }_{\alpha },{\pi }_{\alpha })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)>h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right),\\ \hfill ({\mu }_{\beta },{\pi }_{\beta })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)<h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right),\\ \hfill ({\mu }_{\alpha },{\pi }_{\alpha })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)=h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right)& and\hfill & \hfill H_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)\ge H_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right),\\ \hfill ({\mu }_{\beta },{\pi }_{\beta })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)=h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right)& and\hfill & \hfill H_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)<H_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right).\end{array}\right.$$

and $({\mu }_{\omega },{\pi }_{\omega })$ is defined by

$$({\mu }_{\omega },{\pi }_{\omega })\left(x\right)=\left\{\begin{array}{ccc}\hfill ({\mu }_{\beta },{\pi }_{\beta })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)>h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right),\\ \hfill ({\mu }_{\alpha },{\pi }_{\alpha })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)<h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right),\\ \hfill ({\mu }_{\beta },{\pi }_{\beta })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)=h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right)& and\hfill & \hfill H_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)\ge H_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right),\\ \hfill ({\mu }_{\alpha },{\pi }_{\alpha })\left(x\right)& ,if\hfill & \hfill h_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)=h_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right)& and\hfill & \hfill H_{({\mu }_{\alpha },{\pi }_{\alpha })}\left(x\right)<H_{({\mu }_{\beta },{\pi }_{\beta })}\left(x\right).\end{array}\right.$$

Definition 9.

Let E be a finite set and $\psi \subseteq IFS\left(E\right)$ be a nonempty family of fuzzy sets. The pair $(E,\psi )$ is called a $G-V$$IFM$ on E if it satisfies the following conditions:

1.

If $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi ,({\mu }_{\beta },{\pi }_{\beta })\in IFS\left(E\right)$, and $({\mu }_{\beta },{\pi }_{\beta })\prec ({\mu }_{\alpha },{\pi }_{\alpha })$, then $({\mu }_{\beta },{\pi }_{\beta })\in \psi$.

2.

If $({\mu }_{\alpha },{\pi }_{\alpha }),({\mu }_{\beta },{\pi }_{\beta })\in \psi$, and |supp$({\mu }_{\alpha },{\pi }_{\alpha })|<|$supp$({\mu }_{\beta },{\pi }_{\beta })|$, then there exists $({\mu }_{\omega },{\pi }_{\omega })\in \psi$, such that:

(a) 

$({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\omega },{\pi }_{\omega })⪯({\mu }_{\alpha },{\pi }_{\alpha })\vee ({\mu }_{\beta },{\pi }_{\beta })$;

(b) 

$m({\mu }_{\omega },{\pi }_{\omega })\ge min\{m({\mu }_{\alpha },{\pi }_{\alpha }),m({\mu }_{\beta },{\pi }_{\beta })\}$.

Suppose that $(E,\psi )$ is a $G-V$$IFM$. $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ is called an independent $IFS$ and $\psi$ is called the set of independent $IFSs$. $({\mu }_{\beta },{\pi }_{\beta })\notin \psi$ is called a dependent $IFS$.

If for any $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)$ and for any $x\in E$, ${\pi }_{\alpha }\left(x\right)=0$, then $IFS\left(E\right)$ is actually $FS\left(E\right)$. Thus, $(E,\psi )$ is reduced to $G-V$$FM$.

Theorem 2.

Let $(E,\psi )$ be a $G-V$$IFM$. For each r, $0\le r\le 1$, let

$$I_r=\left\{C_r({\mu }_{\alpha },{\pi }_{\alpha })\right|({\mu }_{\alpha },{\pi }_{\alpha })\in \psi \}$$

Then for each r, $0<r\le 1$,

$$M_r=(E,I_r)$$

is a matroid.

Theorem 3.

Let$(E,\psi )$be a$G-V$$IFM$. Let$M_r=(E,I_r)$be a matroid on E defined in Theorem 2, where$0<r\le 1$. Then there is a finite sequence$r_0<r_1<\cdots <r_n$such that:

(i) 

$r_0=0$,$r_n=1$.

(ii) 

$I_s\ne \left\{\varphi \right\}$ if $0<s\le r_n$,$I_s=\left\{\varphi \right\}$ if $s>r_n$.

(iii) 

If $r_i<s,t<r_{i+1}$, then $I_s=I_t$, where $0\le i\le n-1$.

(iv) 

If $r_i<s<r_{i+1}<t<r_{i+2}$, then $I_s\subset I_t$, where $0\le i\le n-2$.

Then the sequence $r_0,r_1,r_2,\cdots ,r_n$ is called the fundamental sequence of $(E,\psi )$. Moreover, if for any i, $1\le i\le n$, let ${\overline{r}}_i=\frac{r_{i-1}}{r_i}$, then we can get a sequence of matroids $M_{{\overline{r}}_n}\subset M_{{\overline{r}}_{n-1}}\subset \cdots \subset M_{\overline{r_2}}\subset M_{\overline{r_1}}$ which is called the $induced$ matroid sequence.

Note that $C_r({\mu }_{\alpha },{\pi }_{\alpha })=\{x\in E|h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))\ge r\}$, where $0<r\le 1$, and $I_r=\left\{C_r({\mu }_{\alpha },{\pi }_{\alpha })\right|({\mu }_{\alpha },{\pi }_{\alpha })\in \psi \}$, so $I_s=\left\{\varphi \right\}$ not but $I_s=\left\{\varphi \right\}$ when $s>r_n$.

A matroid sequence can be constructed from a $G-V$$IFM$ above. On the contrary, a $G-V$$IFM$ can be constructed from a matroid sequence.

Theorem 4.

Let $0=s_0<s_1<s_2<\cdots <s_n\le 1$ be a finite sequence. Suppose that $M_{s_1},M_{s_2},\cdots ,M_{s_{n-1}},M_{s_n}$ ($M_{s_i}=(E,I_{s_i}),1\le i\le n$) is a matroid sequence on a finite set E and satisfies $I_{s_{i+1}}\subset I_{s_i}$ ($0\le i\le n-1)$. For any $0\le s\le 1$, let

$$I_s=\left\{\begin{array}{ccc}\hfill I_{s_i}& ,if\hfill & \hfill s_{i-1}<s\le s_i,0\le i\le n,\\ \hfill \left\{\varphi \right\}& ,if\hfill & \hfill s_n<s\le 1.\end{array}\right.$$

and let

$${\psi }^{*}=\{({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)|C_s({\mu }_{\alpha },{\pi }_{\alpha })\in I_s,0<s\le 1\}.$$

Then $(E,{\psi }^{*})$ is a $G-V$$IFM$ and its $induced$ matroid sequence is $M_{s_n}\subset M_{s_{n-1}}\subset \cdots \subset M_{s_2}\subset M_{s_1}$, where for $1\le i\le n$, $M_{s_i}=(E,I_{s_i})$.

Theorem 5.

Let $(E,\psi )$ be a $G-V$$IFM$, and for each r, let $0<r\le 1$, $M_r=(E,I_r)$ be a matroid defined by Theorem 2. Let ${\psi }^{*}=\{({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)|C_r({\mu }_{\alpha },{\pi }_{\alpha })\in I_r,0<r\le 1\}$. Then $\psi ={\psi }^{*}$.

Theorem 6.

Let $(E,\psi )$ be a $G-V$$IFM$ and $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)$. Then $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ if and only if $C_{\lambda }({\mu }_{\alpha },{\pi }_{\alpha })\in I_{\lambda }$ for each $\lambda \in R^{+}({\mu }_{\alpha },{\pi }_{\alpha })$.

Theorem 7.

Suppose that $(E,\psi )$ is a $G-V$$IFM$ with the fundamental sequence $0=r_0<r_1<r_2<\cdots <r_n\le 1$. If $I_r=I_{r_i}$ for any $r_{i-1}<r\le r_i$ ($0\le i\le n$), then $(E,\psi )$ is called a closed $G-V$$IFM$.

3. Bases of $\mathit{G}-\mathit{V}$ $\mathit{IFMs}$

Based on $G-V$$IFMs$ and bases of matroids or fuzzy matroids, we propose the concept of the intuitionistic fuzzy basis of a $G-V$$IFM$.

Definition 10.

Let $(E,\psi )$ be a $G-V$$IFM$. $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ is said to be maximal in ψ. If for any $({\mu }_{\beta },{\pi }_{\beta })\in \psi$ and $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })$, then $({\mu }_{\alpha },{\pi }_{\alpha })=({\mu }_{\beta },{\pi }_{\beta })$. I.e., there does not exist $({\mu }_{\beta },{\pi }_{\beta })\in \psi$ such that $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\beta },{\pi }_{\beta })$).

An intuitionistic fuzzy basis ($IFB$ for short) of a $G-V$$IFM$$(E,\psi )$ is a maximal member $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$.

Suppose that $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ and $({\mu }_{\beta },{\pi }_{\beta })\in IFS\left(E\right)$. Let $h({\mu }_{\beta },{\pi }_{\beta })=h({\mu }_{\alpha },{\pi }_{\alpha })$ and $H({\mu }_{\beta },{\pi }_{\beta })<H({\mu }_{\alpha },{\pi }_{\alpha })$; then $({\mu }_{\beta },{\pi }_{\beta })⪯({\mu }_{\alpha },{\pi }_{\alpha })$ and $|({\mu }_{\beta },{\pi }_{\beta })|=|({\mu }_{\alpha },{\pi }_{\alpha })|$. Obviously, $({\mu }_{\beta },{\pi }_{\beta })\in \psi$. Therefore, $({\mu }_{\beta },{\pi }_{\beta })$ here is called an intuitionistic fuzzy sub-basis ($IFSB$ for short) with respect to $IFB$$({\mu }_{\alpha },{\pi }_{\alpha })$ for a $G-V$$IFM$$(E,\psi )$. Generally, there are infinite $IFSBs$ for a $G-V$$IFM$ and their cardinality is the same as that of the corresponding $IFB$.

Definition 11.

An $IFS$$({\mu }_{\alpha },{\pi }_{\alpha })$ is an elementary $IFS$ if $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })=1$. If $({\mu }_{\alpha },{\pi }_{\alpha })$ is an elementary $IFS$ with $A=$supp$({\mu }_{\alpha },{\pi }_{\alpha })$ and $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })=\left\{r\right\}$, then $({\mu }_{\alpha },{\pi }_{\alpha })$ is denoted by $\omega (A,r)$ with support A and height r.

Theorem 8.

Suppose that $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ is an $IFB$ of a $G-V$$IFM$$(E,\psi )$, then ${\pi }_{\alpha }\left(x\right)=0$ for each $x\in E$.

Proof. 

Assume that there exists an $x_0\in E$ such that ${\pi }_{\alpha }\left(x_0\right)=\eta >0$. Let $h({\mu }_{\beta },{\pi }_{\beta })=h({\mu }_{\alpha },{\pi }_{\alpha })$ for each $x\in E$ and

$${\pi }_{\beta }\left(x\right)=\left\{\begin{array}{ccc}\hfill {\pi }_{\alpha }\left(x\right)& ,if\hfill & \hfill x\in Eandx\ne x_0,\\ \hfill \frac{\eta }{2}& ,if\hfill & \hfill x=x_0.\end{array}\right.$$

Then $H({\mu }_{\alpha },{\pi }_{\alpha })<H({\mu }_{\beta },{\pi }_{\beta })$. It follows that $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })$. However, for each $\lambda \in R^{+}({\mu }_{\alpha },{\pi }_{\alpha })$, we have $C_{\lambda }({\mu }_{\beta },{\pi }_{\beta })=C_{\lambda }({\mu }_{\alpha },{\pi }_{\alpha })\in I_{\lambda }$. Then $({\mu }_{\beta },{\pi }_{\beta })\in \psi$ from Theorem 6. This contradicts the hypothesis.

Here, we will use Theorem 6 to prove the next theorem. □

Theorem 9.

Let $(E,\psi )$ be a $G-V$$IFM$ with the fundamental sequence $0=r_0<r_1<r_2<\cdots <r_n\le 1$ and suppose $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $(E,\psi )$; then

$$R^{+}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq \{r_1,r_2,\cdots ,r_n\}.$$

Proof. 

Let $({\mu }_{\alpha },{\pi }_{\alpha })$ be an $IFB$ of $(E,\psi )$. Then $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$. It follows that $C_{\lambda }({\mu }_{\alpha },{\pi }_{\alpha })\in I_{\lambda }$ for each $\lambda \in R^{+}({\mu }_{\alpha },{\pi }_{\alpha })$.

Assume that there is an $s\in R^{+}({\mu }_{\alpha },{\pi }_{\alpha })$ such that $r_i<s<r_{i+1}$. We take $\epsilon =(r_{i+1}-s)/2$ and let $({\mu }_{\beta },{\pi }_{\beta })$ be the elementary $IFS$ which is defined by supp$({\mu }_{\beta },{\pi }_{\beta })=C_s({\mu }_{\alpha },{\pi }_{\alpha })$ and $R^{+}({\mu }_{\beta },{\pi }_{\beta })=s+\epsilon$.

If we let $({\mu }_{\omega },{\pi }_{\omega })=({\mu }_{\alpha },{\pi }_{\alpha })\vee ({\mu }_{\beta },{\pi }_{\beta })$, then for each $r\in (0,1]$, we have

$$C_r({\mu }_{\omega },{\pi }_{\omega })=\left\{\begin{array}{cc}\hfill C_s({\mu }_{\alpha },{\pi }_{\alpha })\in I_s& ,ifr\in (s,s+\epsilon ],\hfill \\ \hfill C_r({\mu }_{\alpha },{\pi }_{\alpha })\in I_r& ,ifr\notin (s,s+\epsilon ].\hfill \end{array}\right.$$

By Theorem 6, we have $({\mu }_{\omega },{\pi }_{\omega })\in \psi$.

By the hypothesis, for $({\mu }_{\alpha },{\pi }_{\alpha })$, we have that there exists $x_0\in supp({\mu }_{\alpha },{\pi }_{\alpha })$ such that $h({\mu }_{\alpha }\left(x_0\right),{\pi }_{\alpha }\left(x_0\right))=s$. Thus $h({\mu }_{\omega }\left(x_0\right),{\pi }_{\omega }\left(x_0\right))=s+\epsilon$. Since $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\omega },{\pi }_{\omega })$, $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\omega },{\pi }_{\omega })$. This contradicts that $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$. □

Theorem 10.

Suppose that $(E,\psi )$ is a $G-V$$IFM$ and $0=r_0<r_1<r_2<\cdots <r_n\le 1$ is the fundamental sequence of $(E,\psi )$. Then $(E,\psi )$ is closed if and only if for any $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$, there exists an $IFB$$({\mu }_{\beta },{\pi }_{\beta })\in \psi$ such that $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })$.

Proof. 

Assume that for any $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$, there exists an $IFB$$({\mu }_{\beta },{\pi }_{\beta })\in \psi$ such that $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })$. If $(E,\psi )$ is not closed.

Let $i_0$ be a positive integer such that if $r_{i_0-1}<t<r_{i_0}$, then $I_{r_{i_0}}\subset I_t$, where

$$I_r=\left\{C_r({\mu }_{\alpha },{\pi }_{\alpha })\right|({\mu }_{\alpha },{\pi }_{\alpha })\in \psi \}.$$

Let A be a basis of $I_t$ but not a basis of $I_{r_{i_0}}$. Let $\omega (A,t)$ be the elementary $IFS$. Obviously, $C_r\left(\omega (A,t)\right)=A\in I_r$ for any $r\in (0,t]$, and $C_r\left(\omega (A,t)\right)=\varnothing \in I_r$ for any $r\in (t,1]$. It follows that $\omega (A,t)\in \psi$.

Suppose that $({\mu }_{\beta },{\pi }_{\beta })\in \psi$ is an $IFB$ and $\omega (A,t)⪯({\mu }_{\beta },{\pi }_{\beta })$. Then $C_t({\mu }_{\beta },{\pi }_{\beta })\in I_t$ and $A\subseteq C_t({\mu }_{\beta },{\pi }_{\beta })$. Since A is a crisp basis of $I_t$, $A=C_t({\mu }_{\beta },{\pi }_{\beta })$.

Since $I_{r_{i_0}}\subseteq I_t$ and by Theorem 5 and Theorem 9, we have

$$A=C_t({\mu }_{\beta },{\pi }_{\beta })=C_{r_{i_0}}({\mu }_{\beta },{\pi }_{\beta })\in I_{r_{i_0}}.$$

Conversely, suppose that $(E,\psi )$ is closed. Let $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ and $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })=\{t_1,t_2,\cdots ,t_k\}$, where $t_1<t_2<\cdots <t_k$. Let $\{t_{p_1},t_{p_2},\cdots ,t_{p_s}\}$ be a subsequence of $\{t_1,t_2,\cdots ,t_k\}$ and $\{r_{q_1},r_{q_2},\cdots ,r_{q_s}\}$ be a subsequence of $\{r_0,r_1,\cdots ,r_n\}$ such that $t_{p_1}=t_1$, and for a given $t_{p_j}$, there is $r_{q_j}=min\left\{r_i\right|r_i\ge t_{p_j}\}$, and for a given $r_{q_j}$ there is $t_{p_{j+1}}=min\left\{t_i\right|t_i>r_{q_j}\}$. It follows that

(1)

$t_{p_1}=t_1$;

(2)

$r_{q_{j-1}}<t_{p_j}<r_{q_j},j=1,2,\cdots ,s$;

(3)

If $t_{p_j}<t_i<t_{p_{j+1}}$, then $r_{q_{j-1}}<t_i\le r_{q_j}$;

(4)

If $t_{p_s}>t_i$, then $r_{q_{s-1}}<t_i\le r_{q_s}$.

Let $A_n\subseteq A_{n-1}\subseteq \cdots \subseteq A_1$ be a nested sequence such that

(a)

supp$({\mu }_{\alpha },{\pi }_{\alpha })\subseteq A_1$, where $A_1$ is a basis of $(E,I_{r_1})$;

(b)

For $i\ge 2$ (where i is an integer), we have $q_{j-1}<i<q_j(q_0=0)$ and $A_i$ is a maximal subset of $A_{i-1}$ in $I_{r_i}$ that contains $C_{t_{p_j}}({\mu }_{\alpha },{\pi }_{\alpha })$;

(c)

For $q_t<i\le n$ (where i is an integer), we have A is a maximal subset of $A_{i-1}$ in $I_{r_i}$.

Let $({\mu }_{{\beta }_i},{\pi }_{{\beta }_i})$ be the elementary $IFS$$\omega (A_i,r_i)$, where $i\in [1,n]$. If $({\mu }_{\beta },{\pi }_{\beta })={\bigvee }_{i=1}^n({\mu }_{{\beta }_i},{\pi }_{{\beta }_i})$, then we can easily get $C_r({\mu }_{\beta },{\pi }_{\beta })\in I_r,r\in (0,1]$. By Theorem 6, $({\mu }_{\beta },{\pi }_{\beta })\in \psi$. From the construction of $({\mu }_{\beta },{\pi }_{\beta })$, $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })\in \psi$ and $({\mu }_{\beta },{\pi }_{\beta })$ is an $IFB$ for $(E,\psi )$, the conclusion is established. □

4. The Judgement of an $\mathit{IFB}$ for a $\mathit{G}-\mathit{V}$ $\mathit{IFM}$

From the proof of Theorem 10, we can get the following result.

Theorem 11.

Suppose that $(E,\psi )$ is a closed $G-V$$IFM$ with the fundamental sequence $0=r_0<r_1<r_2<\cdots <r_n\le 1$ and the induced matroid sequence $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$, where $M_{r_i}=(E,I_{r_i})$($1\le i\le n$). Let $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)$. If $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $(E,\psi )$, then supp$({\mu }_{\alpha },{\pi }_{\alpha })=C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha })$ is a basis of matroid $(E,I_{r_1})$.

Proof. 

Suppose that $m({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $IFM$. Then $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq \{r_1,r_2,\cdots ,r_n\}$ and $C_r({\mu }_{\alpha },{\pi }_{\alpha })\in I_r$ for any $r\in R^{+}({\mu }_{\alpha },{\pi }_{\alpha })$.

Assume that supp$({\mu }_{\alpha },{\pi }_{\alpha })=C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha })$ is not a basis of matroid $(E,I_{r_1})$; then there exists a basis A of $(E,I_{r_1})$ such that supp$({\mu }_{\alpha },{\pi }_{\alpha })=C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha })\subset A$. Let

$$h({\mu }_{\omega }\left(x\right),{\pi }_{\omega }\left(x\right))=\left\{\begin{array}{ccc}\hfill r_1& ,if\hfill & \hfill x\in A\backslash C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha }),\\ \hfill h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))& ,if\hfill & \hfill x\in C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha }),\\ \hfill 0& ,\hfill & \hfill otherwise.\end{array}\right.$$

Then $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\omega },{\pi }_{\omega })$, $R^{+}({\mu }_{\omega },{\pi }_{\omega })\subseteq \{r_1,r_2,\cdots ,r_n\}$ and $C_{r_1}({\mu }_{\omega },{\pi }_{\omega })=A\in I_{r_1}$. Thus, for any $r_1<r<m({\mu }_{\alpha },{\pi }_{\alpha })$,

$$C_r({\mu }_{\omega },{\pi }_{\omega })=C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha })\in I_{m({\mu }_{\alpha },{\pi }_{\alpha })}\subseteq I_r,$$

and for any $m({\mu }_{\alpha },{\pi }_{\alpha })\le r\le 1$,

$$C_r({\mu }_{\omega },{\pi }_{\omega })=C_r({\mu }_{\alpha },{\pi }_{\alpha })\in I_r.$$

From Theorem 6, it follows that $({\mu }_{\omega },{\pi }_{\omega })\in \psi$. Since $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\omega },{\pi }_{\omega })$, it contradicts that $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $IFM$. □

The following necessary and sufficient condition can be used to judge whether a fuzzy set is a fuzzy basis.

Theorem 12

([25]). Let $(E,\psi )$ be a closed $G-V$ fuzzy matroid on E and $0=r_0<r_1<\cdots <r_n\le 1$ be the fundamental sequence. Let $\mu \in FS\left(E\right)$. Suppose $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$ is the induced matroid sequence (where $M_{r_i}=(E,I_{r_i})$,$i=1,2,\cdots ,n$). Then μ is a fuzzy basis of $(E,\psi )$ if and only if μ satisfies:

(i) 

$A_1$=suppμ is a basis of matroid $(E,I_{r_1})$.

(ii) 

There exists a sequence $A_2,\cdots ,A_{n-1},A_n$ ($A_i\in I_{r_i}$) which satisfies $A_i$ is a maximal subset of $A_{i-1}$ in $I_{r_i}$($i=2,3,4,\cdots ,n$) and $A_1\supseteq A_2\supseteq \cdots \supseteq A_{n-1}\supseteq A_n$ such that for any $x\in A_n$, $\mu \left(x\right)=r_n$ and for any $x\in A_i\backslash A_{i+1}$, $\mu \left(x\right)=r_i$, where $i=1,2,3,\cdots ,n-1$.

This result can be extended to intuitionistic fuzzy sets.

Theorem 13.

Suppose $(E,\psi )$ is a closed $G-V$$IFM$ with the fundamental sequence $0=r_0<r_1<r_2<\cdots <r_n\le 1$ and the induced matroid sequence $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$, where $M_{r_i}=(E,I_{r_i})$ ($1\le i\le n$). Let $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)$; then $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $(E,\psi )$ if and only if $({\mu }_{\alpha },{\pi }_{\alpha })$ satisfies:

(I) 

${\pi }_{\alpha }\left(x\right)=0$ for each $x\in E$;

(II) 

The set $A_1=supp({\mu }_{\alpha },{\pi }_{\alpha })$ is a crisp basis of matroid $(E,I_{r_1})$;

(III) 

There exists a sequence $A_2,\cdots ,A_{n-1},A_n$ ($A_i\in I_{r_i}$) which satisfies $A_i$ is a maximal subset of $A_{i-1}$ in $I_{r_i}$ ($i=2,3,\cdots ,n$) and $A_1\supseteq A_2\supseteq \cdots \supseteq A_{n-1}\supseteq A_n$ such that for any $x\in A_n$, $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=r_n$, and for any $x\in A_i\backslash A_{i+1}$ ($i=1,2,\cdots ,n-1$), $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=r_i$.

Proof. 

By Theorem 8 and Theorem 11, we have

(I)

${\pi }_{\alpha }\left(x\right)=0$ for each $x\in E$;

(II)

The set $A_1$=supp$({\mu }_{\alpha },{\pi }_{\alpha })$ is a basis of matroid $(E,I_{r_1})$.

Now we just prove that (III) holds.

Let $A_i=C_{r_i}({\mu }_{\alpha },{\pi }_{\alpha })$ ($2\le i\le n$). By the hypothesis, we have $C_{r_n}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq C_{r_{n-1}}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq \cdots \subseteq C_{r_2}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq C_{r_1}({\mu }_{\alpha },{\pi }_{\alpha })$, That is $A_n\subseteq A_{n-1}\subseteq \cdots \subseteq A_2\subseteq A_1$.

Next, we will prove $A_i$ is a maximal subset of $A_{i-1}$ in $I_{r_i}$, where $k+1\le i\le n$.

Note that $A_1$=supp$({\mu }_{\alpha },{\pi }_{\alpha })$ is the basis of $(E,I_{r_1})$.

Assume that there exists $A_i\in I_{r_i}$ ($2\le i\le n$) such that $A_i$ is not a maximal subset of $A_{i-1}$ in $I_{r_{i-1}}$. Then there is $B\in I_{r_i}$ such that $A_i\subset B$ and B is a maximal subset of $A_{i-1}$.

Let $({\mu }_{\beta },{\pi }_{\beta })\in IFS\left(E\right)$ and ${\pi }_{\beta }\left(x\right)=0$ for each $x\in E$, and if $i=2$, let

$$h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))=\left\{\begin{array}{ccc}\hfill r_1& ,\hfill & \hfill x\in A_1\backslash B,\\ \hfill r_2& ,\hfill & \hfill x\in B\backslash A_2,\\ \hfill h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))& ,\hfill & \hfill x\in A_2.\end{array}\right.$$

If $3\le i\le n$, let

$$h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))=\left\{\begin{array}{ccc}\hfill r_j& ,\hfill & \hfill x\in A_j\backslash A_{j+1},\\ \hfill r_{i-1}& ,\hfill & \hfill x\in A_{i-1}\backslash B,\\ \hfill r_i& ,\hfill & \hfill x\in B\backslash A_i,\\ \hfill h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))& ,\hfill & \hfill x\in A_i.\end{array}\right.$$

where $j=1,2,\cdots ,i-2$. Then $({\mu }_{\alpha },{\pi }_{\alpha })⪯({\mu }_{\beta },{\pi }_{\beta })$. Since $C_{r_i}({\mu }_{\beta },{\pi }_{\beta })=B\in I_{r_i}$, it follows that $C_{r_j}({\mu }_{\beta },{\pi }_{\beta })=A_j\in I_{r_j}$, for any $1\le j\le i-1$, and $C_{r_j}({\mu }_{\beta },{\pi }_{\beta })=C_{r_j}({\mu }_{\alpha },{\pi }_{\alpha })\in I_{r_j}$ for any $i+1\le j\le n$. Then, by Theorem 6, $({\mu }_{\beta },{\pi }_{\beta })\in \psi$, which contradicts that $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $(E,\psi )$.

Conversely, from condition (II) (III), $A_1=$supp$({\mu }_{\alpha },{\pi }_{\alpha })$ is a crisp basis of matroid $(E,I_{r_1})$, $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq \{r_1,r_2,\cdots ,r_n\}$ and $C_{r_i}({\mu }_{\alpha },{\pi }_{\alpha })=A_i\in I_{r_i}$ for any $r_i\in R^{+}({\mu }_{\alpha },{\pi }_{\alpha })$ ($i=1,2,\cdots ,n$). It follows that $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ from Theorem 6. □

$({\mu }_{\alpha },{\pi }_{\alpha })$ is not an $IFB$ of $(E,\psi )$. Since $({\mu }_{\alpha },{\pi }_{\alpha })\in \psi$ and $(E,\psi )$ is a closed $IFM$, there exists an $IFB$$({\mu }_{\beta },{\pi }_{\beta })$ of $(E,\psi )$ such that $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\beta },{\pi }_{\beta })$, so $m({\mu }_{\alpha },{\pi }_{\alpha })\le m({\mu }_{\beta },{\pi }_{\beta })$ and supp$({\mu }_{\alpha },{\pi }_{\alpha })\subseteq$supp$({\mu }_{\beta },{\pi }_{\beta })$.

Case 1. supp$({\mu }_{\alpha },{\pi }_{\alpha })=$ supp$({\mu }_{\beta },{\pi }_{\beta })$. Since $({\mu }_{\beta },{\pi }_{\beta })$ is an $IFB$ of $(E,\psi )$, then ${\pi }_{\beta }\left(x\right)=0$ for each $x\in E$ and $A_1=$supp$({\mu }_{\alpha },{\pi }_{\alpha })=$supp$({\mu }_{\beta },{\pi }_{\beta })$ is a basis of matroid $(E,I_{r_1})$. As $A_1\supseteq A_2\supseteq \cdots \supseteq A_{n-1}\supseteq A_n$ and $A_i$ is a maximal subset of $A_{i-1}$, where $A_i\in I_{r_i}$ ($i=2,3,\cdots ,n$), for any $x\in A_n$, $h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))=r_n$ and for any $x\in A_i\backslash A_{i+1}$ ($i=1,2,\cdots ,n-1$), $h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))=r_i$, for any $x\in$ supp$({\mu }_{\alpha },{\pi }_{\alpha })=$supp$({\mu }_{\beta },{\pi }_{\beta })$, we have $h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=h({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$. Since ${\pi }_{\alpha }\left(x\right)={\pi }_{\beta }\left(x\right)=0$ for each $x\in E$, $H({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))=H({\mu }_{\beta }\left(x\right),{\pi }_{\beta }\left(x\right))$. It follows that $({\mu }_{\alpha },{\pi }_{\alpha })=({\mu }_{\beta },{\pi }_{\beta })$, which contradicts that $({\mu }_{\alpha },{\pi }_{\alpha })\prec ({\mu }_{\beta },{\pi }_{\beta })$, $m({\mu }_{\alpha },{\pi }_{\alpha })\le m({\mu }_{\beta },{\pi }_{\beta })$.

Case 2. supp$({\mu }_{\alpha },{\pi }_{\alpha })\subset$ supp$({\mu }_{\beta },{\pi }_{\beta })$. Since $({\mu }_{\beta },{\pi }_{\beta })$ is an $IFB$ of $(E,\psi )$, $C_{m({\mu }_{\beta },{\pi }_{\beta })}({\mu }_{\beta },{\pi }_{\beta })=$ supp$({\mu }_{\beta },{\pi }_{\beta })$ is a basis of matroid $(E,I_{r_1})$. From condition (II), $C_{m({\mu }_{\alpha },{\pi }_{\alpha })}({\mu }_{\alpha },{\pi }_{\alpha })=$supp$({\mu }_{\alpha },{\pi }_{\alpha })$ is also a basis of matroid $(E,I_{r_1})$. Then supp$({\mu }_{\alpha },{\pi }_{\alpha })=$supp$({\mu }_{\beta },{\pi }_{\beta })$, which is in contradiction with supp$({\mu }_{\alpha },{\pi }_{\alpha })\subset$supp$({\mu }_{\beta },{\pi }_{\beta })$.

Therefore, $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $(E,\psi )$.

The following corollary is obvious.

Corollary 1.

Suppose $(E,\psi )$ is a closed $G-V$$IFM$ with the fundamental sequence $0=r_0<r_1<r_2<\cdots <r_n\le 1$ and the induced matroid sequence $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$, where $M_{r_i}=(E,I_{r_i})$ ($1\le i\le n$). Let $({\mu }_{\alpha },0)\in IFS\left(E\right)$. Then $({\mu }_{\alpha },0)$ is an $IFB$ of $(E,\psi )$ if and only if the $IFS$$({\mu }_{\alpha },0)$ satisfies:

(1) 

$A_1$ is a crisp basis of $(E,I_{r_1})$, where $A_1=supp({\mu }_{\alpha },0)$.

(2) 

There exist $A_2,\cdots ,A_{n-1},A_n$ ($A_i\in I_{r_i}$) which satisfy $A_1\supseteq A_2\supseteq \cdots \supseteq A_{n-1}\supseteq A_n$ and $A_i$ is a maximal subset of $A_{i-1}$ ($i=2,3,\cdots ,n$) such that $h({\mu }_{\alpha }\left(x\right),0)={\mu }_{\alpha }\left(x\right)=r_n$ for any $x\in A_n$, and $h({\mu }_{\alpha }\left(x\right),0)={\mu }_{\alpha }\left(x\right)=r_i$ for any $x\in A_i\backslash A_{i+1}$, $i=1,2,\cdots ,n-1$.

Theorem 14.

Let E be a finite set. Suppose that there is the same fundamental sequence $0=r_0<r_1<r_2<\cdots <r_n\le 1$ and the same induced matroid sequence $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$ for $G-V$ fuzzy matroid $(E,\overline{\psi })$ and $G-V$$IFM$$(E,\psi )$, where $M_{r_i}=(E,I_{r_i})$ ($i=1,2,\cdots ,n-1$). Then ${\mu }_{\alpha }\in FS\left(E\right)$ is a fuzzy basis of $FM=(E,\overline{\psi })$ if and only if $({\mu }_{\alpha },0)\in IFS\left(E\right)$ is an $IFB$ of $(E,\psi )$.

Proof. 

By the hypothesis and Theorem 12, we have ${\mu }_{\alpha }$ is a fuzzy basis of $(E,\overline{\psi })$ if and only if the fuzzy set ${\mu }_{\alpha }$ satisfies:

(1)

$A_1$ is a basis of $(E,I_{r_1})$, where $A_1$=supp${\mu }_{\alpha }$.

(2)

There exist $A_2,\cdots ,A_{n-1},A_n$ which satisfy $A_i$ is a maximal subset of $A_{i-1}$ ($i=2,3,\cdots ,n$) and $A_1\supseteq A_2\supseteq \cdots \supseteq A_{n-1}\supseteq A_n$ such that for any $x\in A_n$, ${\mu }_{\alpha }\left(x\right)=r_n$, and for any $x\in A_i\backslash A_{i+1}$$(i=1,2,\cdots ,n-1),$${\mu }_{\alpha }\left(x\right)=r_i$.

These two conditions hold if and only if $({\mu }_{\alpha },0)$ satisfies:

(1)

$A_1=supp({\mu }_{\alpha },0)$ is a crisp basis of matroid $(E,I_{r_1})$.

(2)

For the above $A_i,i=1,2,\cdots ,n$, we have for any $x\in A_n$, $h({\mu }_{\alpha }\left(x\right),0)={\mu }_{\alpha }\left(x\right)=r_n$, and for any $x\in A_i\backslash A_{i+1}$$(i=1,2,\cdots ,n-1),$$h({\mu }_{\alpha }\left(x\right),0)={\mu }_{\alpha }\left(x\right)=r_i$.

□

5. A Tree Structure of a Closed $\mathit{G}-\mathit{V}$ $\mathit{IFM}$

From Theorem 13, a tree structure of a closed $G-V$$IFM$ is proposed below, which is similar to the tree structure introduced in [25].

Let $(E,\psi )$ be a closed $G-V$$IFM$ on E, $0=r_0<r_1<r_2<\cdots <r_n\le 1$ be the fundamental sequence and $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$ be the $IFM$-induced matroid sequence (where $M_{r_i}=(E,I_{r_i})$ ($1\le i\le n$)). Suppose that $({\mu }_{\alpha },{\pi }_{\alpha })$ is an $IFB$ of $(E,\psi )$ and $B_1=$supp$({\mu }_{\alpha },{\pi }_{\alpha })$ is a crisp basis of matroid $(E,I_{r_1})$. Then, from Theorem 13, there exists a sequence $B_{2,1},\cdots ,B_{n-1,1},B_{n,1}$ ($B_{i,1}\in I_{r_i}$, $i=2,3,\cdots ,n$) such that $B_{i,1}$ is a maximal subset of $B_{i-1,1}$ ($i=2,3,\cdots ,n$) in $I_{r_i}$ and $B_1\supseteq B_{2,1}\supseteq \cdots \supseteq B_{n-1,1}\supseteq B_{n,1}$. Obviously, $C_{r_i}({\mu }_{\alpha },{\pi }_{\alpha })=B_{i,1}$, $i=1,2,\cdots ,n$. The number of the sequence $B_1,B_{2,1},\cdots ,B_{n-1,1},B_{n,1}$ is determined by the number of the maximal subsets of the previous maximal subset in the next level based on the same $IFB$$({\mu }_{\alpha },{\pi }_{\alpha })$. Obviously, each of the sequence can be constructed a brunch of a tree. All the sequences of the same $IFB$$({\mu }_{\alpha },{\pi }_{\alpha })$ can be constructed a tree. Since there are many $IFBs$, there are many trees which become a forest. The forest is called a tree structure of the closed $G-V$$IFM$$(E,\psi )$ (Figure 1).

Figure 1. The tree structure of a closed G-V $IFM$.

Definition 12.

The set of trees constructed by the sequences in Theorem 13 is the tree structure of a closed $G-V$$IFM$$(E,\psi )$, denoted by $T(E,\psi )$ (T for short) (Figure 1), which is defined below.

Remark 1.

There is one branch corresponding to a leaf in T and vice versa. From Theorem 13 and the construction of T, a branch of T and an $IFB$ of $(E,\psi )$ are one-to-one corresponding. Thus, for $(E,\psi )$, the number of the $IFB$ is equal to the number of leaves $\left(B_{n,j}\right)$ of T.

Example 1.

Let $E=\{a,b,c\}$, $I_1=\{\varnothing ,\left\{a\right\},\left\{b\right\}\}$, $I_{1/3}=\{\varnothing ,\left\{a\right\},\left\{b\right\},\left\{c\right\},\{a,b\},\{a,c\}\}$, $I_{1/5}=\{\varnothing ,\left\{a\right\},\left\{b\right\},\left\{c\right\},\{a,b\},\{a,c\},\{b,c\}\}$. Then $(E,I_1)$, $(E,I_{1/3})$ and $(E,I_{1/5})$ are all matroids, and $I_{1/5}$, $I_{1/3}$, $I_1$. Let

$$I_r=\left\{\begin{array}{cc}\hfill I_{1/5}& ,0<r\le \frac{1}{5},\hfill \\ \hfill I_{1/3}& ,\frac{1}{5}<r\le \frac{1}{3},\hfill \\ \hfill I_1& ,\frac{1}{3}<r\le 1.\hfill \end{array}\right.$$

and let $\psi =\{({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)|C_r({\mu }_{\alpha },{\pi }_{\alpha })\in I_r\}$, where $r\in (0,1]$. From Definition 2.16, $(E,\psi )$ is a closed $G-V$$IFM$. The tree structure T is shown in Figure 2.

Figure 2. The tree structure of Example 5.3.

From Figure 2, there are three trees and five leaves in T. By Remark 1, there are five $IFBs$ of $(E,\psi )$, which are as follows:

$$({\mu }_{{\alpha }_1}\left(x\right),{\pi }_{{\alpha }_1}\left(x\right))=\left\{\begin{array}{cc}\hfill (1,0)& ,x=a,\hfill \\ \hfill (\frac{1}{3},0)& ,x=b,\hfill \\ \hfill (0,0)& ,x=c.\hfill \end{array}\right.$$

$$({\mu }_{{\alpha }_2}\left(x\right),{\pi }_{{\alpha }_2}\left(x\right))=\left\{\begin{array}{cc}\hfill (\frac{1}{3},0)& ,x=a,\hfill \\ \hfill (1,0)& ,x=b,\hfill \\ \hfill (0,0)& ,x=c.\hfill \end{array}\right.$$

$$({\mu }_{{\alpha }_3}\left(x\right),{\pi }_{{\alpha }_3}\left(x\right))=\left\{\begin{array}{cc}\hfill (1,0)& ,x=a,\hfill \\ \hfill (0,0)& ,x=b,\hfill \\ \hfill (\frac{1}{3},0)& ,x=c.\hfill \end{array}\right.$$

$$({\mu }_{{\alpha }_4}\left(x\right),{\pi }_{{\alpha }_4}\left(x\right))=\left\{\begin{array}{cc}\hfill (0,0)& ,x=a,\hfill \\ \hfill (1,0)& ,x=b,\hfill \\ \hfill (\frac{1}{5},0)& ,x=c.\hfill \end{array}\right.$$

$$({\mu }_{{\alpha }_5}\left(x\right),{\pi }_{{\alpha }_5}\left(x\right))=\left\{\begin{array}{cc}\hfill (0,0)& ,x=a,\hfill \\ \hfill (\frac{1}{5},0)& ,x=b,\hfill \\ \hfill (\frac{1}{3},0)& ,x=c.\hfill \end{array}\right.$$

Then the values of the similarity function h for the five $IFBs$ are below:

$$h({\mu }_{{\alpha }_1}\left(x\right),{\pi }_{{\alpha }_1}\left(x\right))=\left\{\begin{array}{cc}\hfill 1& ,x=a,\hfill \\ \hfill \frac{1}{3}& ,x=b,\hfill \\ \hfill 0& ,x=c.\hfill \end{array}\right.$$

$$h({\mu }_{{\alpha }_2}\left(x\right),{\pi }_{{\alpha }_2}\left(x\right))=\left\{\begin{array}{cc}\hfill \frac{1}{3}& ,x=a,\hfill \\ \hfill 1& ,x=b,\hfill \\ \hfill 0& ,x=c.\hfill \end{array}\right.$$

$$h({\mu }_{{\alpha }_3}\left(x\right),{\pi }_{{\alpha }_3}\left(x\right))=\left\{\begin{array}{cc}\hfill 1& ,x=a,\hfill \\ \hfill 0& ,x=b,\hfill \\ \hfill \frac{1}{3}& ,x=c.\hfill \end{array}\right.$$

$$h({\mu }_{{\alpha }_4}\left(x\right),{\pi }_{{\alpha }_4}\left(x\right))=\left\{\begin{array}{cc}\hfill 0& ,x=a,\hfill \\ \hfill 1& ,x=b,\hfill \\ \hfill \frac{1}{5}& ,x=c.\hfill \end{array}\right.$$

$$h({\mu }_{{\alpha }_5}\left(x\right),{\pi }_{{\alpha }_5}\left(x\right))=\left\{\begin{array}{cc}\hfill 0& ,x=a,\hfill \\ \hfill \frac{1}{5}& ,x=b,\hfill \\ \hfill \frac{1}{3}& ,x=c.\hfill \end{array}\right.$$

Next, we discuss the properties of T for $(E,\psi )$.

Theorem 15.

Let $(E,\psi )$ be a closed $G-V$$IFM$ on E, $0=r_0<r_1<\cdots <r_n\le 1$ be the fundamental sequence and $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$ (where $M_{r_i}=(E,I_{r_i})$ ($1\le i\le n$)) be the induced matroid sequence. Let T be the tree structure of $(E,\psi )$. Then each basis $B_i^{k_i}$ of the induced matroid $(E,I_{r_i})$($i=1,2,\cdots ,n.$$k_i$ is a positive integer) is in $r_i$ level of T.

Proof. 

For any i($i=1,2,\cdots ,n$), if $i=1$, since each basis $B_1^{k_1}$ of matriod $M_{r_1}=(E,I_{r_1})$ is the root of each tree in T, $B_1^{k_1}$ is in $r_1$ level.

If $i\ne 1$($i=2,3,\cdots ,n$), for any basis $B_i^{k_i}$ of matroid $M_{r_i}=(E,I_{r_i})$—since $(E,I_{r_i})\subset (E,I_{r_{i-1}})$, it follows that $B_i^{k_i}\in I_{r_{i-1}}$—then there exists a basis $B_{i-1}^{k_{i-1}}$ of $(E,I_{r_{i-1}})$ such that $B_i^{k_i}\subseteq B_{i-1}^{k_{i-1}}$. Obviously, $B_i^{k_i}$ is a maximal subset of $B_{i-1}^{k_{i-1}}$ in $I_{r_i}$. It implies that $B_i^{k_i}$ is in $r_i$ level of T.

Note that the converse of Theorem 15 does not hold. In Example 1, $\{a,b\}$,$\{a,c\}$ are both the bases of matroid $(E,I_{1/3})$ in the second level, but $\left\{b\right\}$,$\left\{c\right\}$ are not the bases. □

Theorem 16.

Let $(E,\psi )$ be a closed $G-V$$IFM$ on E, $0=r_0<r_1<\cdots <r_n\le 1$ be the fundamental sequence and $M_{r_1}\supset M_{r_2}\supset \cdots \supset M_{r_n}$ (where $M_{r_i}=(E,I_{r_i})$ ($1\le i\le n$)) be the induced matroid sequence. Let T be the tree structure of $(E,\psi )$. Suppose that ${\mathbf{B}}_i$ is the collection of the sets in $r_i$ level of T, where $i=1,2,\cdots ,n$. Let $J_{r_i}=\{X\mid X\subseteq B,B\in {\mathbf{B}}_i\}$. Then $J_{r_i}=I_{r_i}$.

Proof. 

For any $Y\in I_{r_i}$, by the hypothesis, there is a basis B of matroid $(E,I_{r_i})$ such that $Y\subseteq B$. By Theorem 15, all bases of $(E,I_{r_i})$ are in $r_i$ level T, where $i=1,2,\cdots ,n$. Then $B\in {\mathbf{B}}_i$. It implies that $Y\in \left\{X\right|X\subseteq B,B\in {\mathbf{B}}_i\}=J_{r_i}$. Thus, $I_{r_i}\subseteq J_{r_i}$.

On the other hand, for any $Y\in J_{r_i}$, there exists a set $B\in {\mathbf{B}}_i$ in $r_i$ ($i=1,2,\cdots ,n$) level of T such that $Y\subseteq B$. By Theorem 13, $B\in I_{r_i}$, $Y\in I_{r_i}$. That implies that $J_{r_i}\subseteq I_{r_i}$.

Therefore, $J_{r_i}=I_{r_i}$. □

Remark 2.

Let $(E,\psi )$ be a closed $G-V$$IFM$ on E and T be its tree structure. Suppose that ${\mathbf{B}}_i$ is the collection of the maximal subsets in $r_i$ level of T. Then the bases of $M_{r_i}=(E,I_{r_i})$ ($i=1,2,\cdots ,n$) belong to ${\mathbf{B}}_i$.

Theorem 17.

Let $(E,\psi )$ be a closed $G-V$$IFM$ on E and T be its tree structure. Suppose that the sequence $B_1,B_2,\cdots ,B_n$ ($B_i$ is in $i-th$ level) of T satisfying $B_n\ne \varnothing$ and $B_1\supset B_2\supset \cdots \supset B_n$. For any $x\in B_n$, let $({\mu }_{\alpha },{\pi }_{\alpha })\in IFS\left(E\right)$ and $k_n=h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$ and for any $x\in B_i\backslash B_{i+1}$ ($i=1,2,\cdots ,n-1$), let $k_i=h({\mu }_{\alpha }\left(x\right),{\pi }_{\alpha }\left(x\right))$. Then $0=k_0,k_1,k_2,\cdots ,k_n$ is the fundamental sequence of $(E,\psi )$.

Proof. 

Let $0=r_0<r_1<r_2<\cdots <r_n\le 1$ be the fundamental sequence of $(E,\psi )$. By the hypothesis and Theorem 13, $({\mu }_{\alpha },{\pi }_{\alpha })$ is a fuzzy basis of $(E,\psi )$. Thus $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })\subseteq \{r_1,r_2,\cdots ,r_n\}$. Suppose that a sequence $B_1,B_2,\cdots ,B_n$ satisfies $B_n\ne \varnothing$ and $B_1\supset B_2\supset \cdots \supset B_n$. It follows that $B_i\backslash B_{i+1}\ne \varnothing$ ($i=1,2,\cdots ,n-1$). Then $k_i=h({\mu }_{\alpha },{\pi }_{\alpha })\ne 0$ for any i ($i=1,2,\cdots ,n-1$) and $R^{+}({\mu }_{\alpha },{\pi }_{\alpha })=\{k_1,k_2,\cdots ,k_n\}$. Thus $\{k_1,k_2,\cdots ,k_n\}\subseteq \{r_1,r_2,\cdots ,r_n\}$. That implies that $\{k_1,k_2,\cdots ,k_n\}=\{r_1,r_2,\cdots ,r_n\}$.

Therefore, $k_0,k_1,k_2,\cdots ,k_n$ is the fundamental sequence of $(E,\psi )$. □

6. Conclusions

In this paper, the $IFB$ of $G-V$$IFMs$ was defined by using the related concept of $G-V$ fuzzy matroids. Some conclusions of $G-V$ fuzzy matroids have been extended to $G-V$$IFMs$. Especially, the judgement of an $IFB$ was presented and proven, and the tree structure of closed $G-V$$IFMs$ and its properties were discussed. We will discuss another important concept and its properties of $G-V$$IFMs$–intuitionistic fuzzy circuits in a subsequent article.