Computation of all generalized inverses of an intuitionistic fuzzy matrix 1

Abstract

The generalized inverse is an important result in matrix theory. In this paper, a necessary and sufficient condition for the regularity of a given intuitionistic fuzzy matrix is provided. For regular intuitionistic fuzzy matrices, how to solve the problem of finding all generalized inverses is discussed.

Keywords

Generalized inverse matrix regular intuitionistic fuzzy matrix intuitionistic fuzzy set

1 Introduction

In 1986, Atanassov introduced intuitionistic fuzzy sets which constitute a generalization of the notion of fuzzy sets [1]. Fuzzy sets give the degree of membership of an element in a given set, while intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership. It is possible to model hesitation and uncertainty by additional degree. This greatly enriched and developed the fuzzy set theory that was introduced by Zadeh in 1965 [2].

Matrices have important roles in various areas of science and engineering. However, classical matrix theory can not solve problems involving various types of uncertainties. These types of problem are solved by using fuzzy matrices. Fuzzy matrices were introduced for the first time in a paper by Thomason [3], in which he discussed the convergence of powers of fuzzy matrices. As a natural generalization of fuzzy matrices, Im and Lee defined the concept of intuitionistic fuzzy matrices (IFMs) [4]. IFMs are widely applied in decision making and control. Pal introduced the intuitionistic fuzzy determinant for the first time [5]. Using the idea of intuitionistic fuzzy sets and the intuitionistic fuzzy determinant, Pal et al. developed IFMs and studied several of their properties in 2002 [6]. Shyamal and Pal also found the distance between IFMs [7]. In the past ten years, researchers have done substantial work on IFMs. Recently, Sriram and Murugadas studied semiring of IFMs [8]. Pradhan and Pal studied linear transformations of intuitionistic fuzzy vectors and the convergence of IFMs with respect to different binary operations [9, 10]. They also investigated convergence of IFMs with respect to maxgeneralized mean-mingeneralized mean powers [11]. For other works on IFMs, see[12 –19].

A generalized inverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. The concept of the generalized inverse was introduced in 1903 by Fredholm [20]. The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the ‘inverse’, in some sense, for a wider class of matrices than just invertible ones. Similar to the inverse of matrices, the generalized inverse of matrices is important in a variety of applications such as robotics, control, signal processing and associative memories.

In the past few decades, scholars have studied the generalized inverse of a matrix over different algebraic structures, such as commutative rings, arbitrary rings and semirings [21 –25]. In this paper, the generalized inverse of IFMs is considered. As a classic research object, Khan and Pal introduced the concept of generalized inverses for IFMs [26]. Pradhan and Pal found the generalized inverse of block IFMs [27]. They also introduced some results on the generalized inverse of IFMs in 2014 [28]. Much work about the generalized inverse of IFMs is still in progress. Here, a naturally arising problem is considered: how should the regularity of an IFM be judged; and. if it is regular, how can its generalized inverses be found? In this paper, an answer to this problem is proposed.

The remainder of this paper is organized as follows. Section 2 presents some fundamental concepts and some background in this area of study. Section 3 is devoted to discussions on the regularity of a given IFM and the calculation of the largest g-inverse of a regular IFM. In Section 4, a method to find all the generalized inverses of a given IFM is presented.

2 Preliminary definitions

In this section, some preliminary definitions regarding the topic are recalled.

Definition 2.1. [1] An intuitionistic fuzzy set (IFS) A on universe X can be defined as follows: $A = {〈 x, μ_{A} (x), λ_{A} (x) 〉 ∣ x \in X}$ where μ_A (x) : X → [0, 1] and λ_A (x) : X → [0, 1] with the property:0 ≤ μ_A (x) + λ_A (x) ≤1 (∀ x ∈ X). The values μ_A (x) and λ_A (x) denotes the degree of membership and non-membership of x to A, respectively. π_A (x) =1 - μ_A (x) - λ_A (x) is called the intuitionistic fuzzy index.

Definition 2.2. [6] An intuitionistic fuzzy matrix (IFM) A of order m × n is defined as A = (a_ij, < μ_{a
_ij}, λ_{a
_ij} >) _m×n where μ_{a
_ij} and λ_{a
_ij} are called membership and non-membership values of a_ij in A, maintaining the condition 0 ≤ μ_{a
_ij} + λ_{a
_ij} ≤ 1. For simplicity, it is possible to write A = (a_ij) _m×n where a_ij = (μ_{a
_ij}, λ_{a
_ij}). Let F_mn denote the set of all IFMs of order m × n. If m = n, in short, F_n, the set of all square IFMs of order n.

Definition 2.3. [6] Let A = ((μ_{a
_ij}, λ_{a
_ij})) ∈ F_mn and B = ((μ_{b
_ij}, λ_{b
_ij})) ∈ F_mn, then the matrix addition is given by $A + B = ((μ_{a_{ij}} \lor μ_{b_{ij}}, λ_{a_{ij}} \land λ_{b_{ij}})) \in F_{mn}$ and the product of A = ((μ_{a
_ij}, λ_{a
_ij})) ∈ F_mp and B = ((μ_{b
_ij}, λ_{b
_ij})) ∈ F_pn is given by, $AB = ((⋁_{k} (μ_{a_{ik}} \land μ_{b_{kj}}), ⋀_{k} (λ_{a_{ik}} \lor λ_{b_{kj}}))$ where k = 1, 2, … p, i = 1, 2, … m and j = 1, 2, … n.

Definition 2.4. If A = (a_ij, < μ_{a
_ij}, λ_{a
_ij} >) and B = (b_ij, < μ_{b
_ij}, λ_{b
_ij} >) are two IFMs of order m × n, then A ⊆ B ⇔ μ_{a
_ij} ≤ μ_{b
_ij} and λ_{a
_ij} ≥ λ_{b
_ij}.

Definition 2.5. [26]. A matrix A ∈ F_mn is said to be regular if there exists X ∈ F_nm such that AXA = A. In this case X is called a generalized inverse (g-inverse) of A and it is denoted by A^-. A {1} denotes the set of all g-inverses of A.

Example 2.6. Consider the IFMs $A = (\begin{matrix} (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) \\ (\begin{matrix} 0.5, & 0.5 \end{matrix}) (\begin{matrix} 0, & 1 \end{matrix}) \end{matrix})$ and $B = (\begin{matrix} (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.6, & 0 \end{matrix}) \\ (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.6, & 0 \end{matrix}) \end{matrix})$

ABA = A. Hence, A is regular, B ∈ A {1}.

Definition 2.7. X ∈ A {1} is called a maximum element of A {1} if Y ⊆ X for all Y ∈ A {1}.

In this paper, the infimum of ∅ is 1 and the supremum of ∅ is 0.

3 The largest g-inverse

Theorem 3.1. If matrix A ∈ F_mn is regular, there exists $\tilde{X} \in F_{nm}$ such that $A \tilde{X} A = A$ , where $\tilde{X} =$ $({\tilde{x}}_{jk}, < μ_{{\tilde{x}}_{jk}}, λ_{{\tilde{x}}_{jk}} >), μ_{{\tilde{x}}_{jk}} = \land {μ_{a_{st}} | μ_{a_{st}} < (μ_{a_{sj}} \land$ $μ_{a_{kt}})}, λ_{{\tilde{x}}_{jk}} = \lor {λ_{a_{st}} | λ_{a_{st}} > (λ_{a_{sj}} \lor λ_{a_{kt}})}, j = 1, 2,$ …, n ; k = 1, 2, …, m. In fact, the above is another necessary and sufficient condition for a given matrix to be regular. $\tilde{X} \in F_{nm}$ is the maximum element of A {1}.

Proof. Let X ∈ A {1}, then AXA = A. For all (i, l) (i = 1, 2, …, m ; l = 1, 2, …, n), $\begin{matrix} ⋁_{k} [(⋁_{j} (μ_{a_{ij}} \land μ_{x_{jk}})) \land μ_{a_{kl}}] \\ = μ_{a_{il}}, ⋀_{k} [(⋀_{j} (λ_{a_{ij}} \lor λ_{x_{jk}})) \lor λ_{a_{kl}}] = λ_{a_{il}} \\ ⋁_{k} ⋁_{j} (μ_{a_{ij}} \land μ_{x_{jk}} \land μ_{a_{kl}}) \\ = μ_{a_{il}}, ⋀_{k} ⋀_{j} (λ_{a_{ij}} \lor λ_{x_{jk}} \lor λ_{a_{kl}}) = λ_{a_{il}} \end{matrix}$

Hence, for all (j, k), $μ_{a_{ij}} \land μ_{x_{jk}} \land μ_{a_{kl}} \leq μ_{a_{il}}, λ_{a_{ij}} \lor λ_{x_{jk}} \lor λ_{a_{kl}} \geq λ_{a_{il}} .$

Therefore, for all (j, k), $\begin{matrix} 0 \leq μ_{x_{jk}} \leq 1 if μ_{a_{ij}} \land μ_{a_{kl}} \leq μ_{a_{il}} and 0 \leq μ_{x_{jk}} \\ \leq μ_{a_{il}} if μ_{a_{ij}} \land μ_{a_{kl}} > μ_{a_{il}} . \end{matrix}$ $\begin{matrix} 0 \leq λ_{x_{jk}} \leq 1 if λ_{a_{ij}} \lor λ_{a_{kl}} \geq λ_{a_{il}} and 0 \leq λ_{a_{il}} \\ \leq λ_{x_{jk}} if λ_{a_{ij}} \lor λ_{a_{kl}} < λ_{a_{il}} . \end{matrix}$

Hence, for all (j, k), $0 \leq μ_{x_{jk}} \leq \land {μ_{a_{il}} | μ_{a_{il}} < (μ_{a_{ij}} \land μ_{a_{kl}})} = μ_{{\tilde{x}}_{jk}}$ and $0 \leq λ_{{\tilde{x}}_{jk}} = \lor {λ_{a_{il}} | λ_{a_{il}} > (λ_{a_{ij}} \lor λ_{a_{kl}})} \leq λ_{x_{jk}} .$

Then $X \subseteq \tilde{X}, A = AXA \subseteq A \tilde{X} A$ .

Conversely, for all i, l and j, k $\begin{matrix} μ_{a_{ij}} \land μ_{{\tilde{x}}_{jk}} \land μ_{a_{kl}} \leq μ_{{\tilde{x}}_{jk}} \\ = \land {μ_{a_{st}} | μ_{a_{st}} < (μ_{a_{sj}} \land μ_{a_{kt}})} \leq μ_{a_{il}}, \\ whenever μ_{a_{ij}} \land μ_{a_{kl}} > μ_{a_{il}} . \end{matrix}$ $\begin{matrix} λ_{a_{ij}} \lor λ_{{\tilde{x}}_{jk}} \lor λ_{a_{kl}} \geq λ_{{\tilde{x}}_{jk}} \\ = \lor {λ_{a_{st}} | λ_{a_{st}} > (λ_{a_{sj}} \lor λ_{a_{kt}})} \geq λ_{a_{il}}, \\ whenever λ_{a_{ij}} \lor λ_{a_{kl}} < λ_{a_{il}} . \end{matrix}$

If μ_{a
_ij} ∧ μ_{a
_kl} ≤ μ_{a
_il} and λ_{a
_ij} ∨ λ_{a
_kl} ≥ λ_{a
_il}, $μ_{a_{ij}} \land μ_{{\tilde{x}}_{jk}} \land μ_{a_{kl}} \leq μ_{a_{ij}} \land μ_{a_{kl}} \leq μ_{a_{il}}$ $λ_{a_{ij}} \lor λ_{{\tilde{x}}_{jk}} \lor λ_{a_{kl}} \geq λ_{a_{ij}} \lor λ_{a_{kl}} \geq λ_{a_{il}}$

Hence, for all (j, k), $μ_{a_{ij}} \land μ_{{\tilde{x}}_{jk}} \land μ_{a_{kl}} \leq μ_{a_{il}}$ , $λ_{a_{ij}} \lor λ_{{\tilde{x}}_{jk}} \lor λ_{a_{kl}} \geq λ_{a_{il}}$

Therefore, for all i, l, $\begin{matrix} ⋁_{j} ⋁_{k} (μ_{a_{ij}} \land μ_{{\tilde{x}}_{jk}} \land μ_{a_{kl}}) \\ \leq μ_{a_{il}}, ⋀_{j} ⋀_{k} (λ_{a_{ij}} \lor λ_{{\tilde{x}}_{jk}} \lor λ_{a_{kl}}) \geq λ_{a_{il}} \end{matrix}$

This shows that $A \supseteq A \tilde{X} A$ . Therefore, $A = A \tilde{X} A$ , $\tilde{X} \in F_{nm}$ is the maximum element of A {1}.

Example 3.2. An intuitionistic fuzzy matrix $A = (\begin{matrix} (\begin{matrix} 0.5, & 0.2 \end{matrix}) (\begin{matrix} 0.5, & 0.2 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.4, & 0.3 \end{matrix}) (\begin{matrix} 0.4, & 0.3 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.5, & 0.2 \end{matrix}) \end{matrix})$

By the formula $\tilde{X} = ({\tilde{x}}_{jk}, < μ_{{\tilde{x}}_{jk}}, λ_{{\tilde{x}}_{jk}} >), μ_{{\tilde{x}}_{jk}} = \land {μ_{a_{st}} | μ_{a_{st}} < (μ_{a_{sj}} \land μ_{a_{kt}})}, λ_{{\tilde{x}}_{jk}} = \lor {λ_{a_{st}} | λ_{a_{st}} > (λ_{a_{sj}} \lor λ_{a_{kt}})}, j = 1, 2, \dots, n; k = 1, 2, \dots, m$ . For the purposes of this work, $μ_{{\tilde{x}}_{jk}}$ and $λ_{{\tilde{x}}_{jk}}$ , respectively. $\begin{matrix} μ_{{\tilde{x}}_{jk}} = \land {μ_{a_{st}} | μ_{a_{11}} < (μ_{a_{1 j}} \land μ_{a_{k 1}}), μ_{a_{12}} \\ < (μ_{a_{1 j}} \land μ_{a_{k 2}}), \dots μ_{a_{1 n}} < (μ_{a_{1 j}} \land μ_{a_{kn}}), \\ μ_{a_{21}} < (μ_{a_{2 j}} \land μ_{a_{k 1}}), μ_{a_{22}} < (μ_{a_{2 j}} \land μ_{a_{k 2}}), \dots μ_{a_{2 n}} \\ < (μ_{a_{2 j}} \land μ_{a_{kn}}), \\ \dots \dots \dots \dots \\ μ_{a_{m 1}} < (μ_{a_{mj}} \land μ_{a_{k 1}}), μ_{a_{m 2}} < (μ_{a_{mj}} \land μ_{a_{k 2}}), \dots μ_{a_{mn}} \\ < (μ_{a_{mj}} \land μ_{a_{kn}})} \end{matrix}$ $\begin{matrix} λ_{{\tilde{x}}_{jk}} = \lor {λ_{a_{st}} | λ_{a_{11}} > (λ_{a_{1 j}} \lor λ_{a_{k 1}}), λ_{a_{12}} \\ > (λ_{a_{1 j}} \lor λ_{a_{k 2}}), \dots λ_{a_{1 n}} > (λ_{a_{1 j}} \lor λ_{a_{kn}}), \end{matrix}$ $\begin{matrix} λ_{a_{21}} > (λ_{a_{2 j}} \lor λ_{a_{k 1}}), λ_{a_{22}} > (λ_{a_{2 j}} \lor λ_{a_{k 2}}), \dots μ_{a_{2 n}} \\ > (μ_{a_{2 j}} \lor μ_{a_{kn}}), \\ \dots \dots \dots \dots \\ λ_{a_{m 1}} > (λ_{a_{mj}} \lor λ_{a_{k 1}}), λ_{a_{m 2}} > (λ_{a_{mj}} \lor λ_{a_{k 2}}), \dots λ_{a_{mn}} \\ > (λ_{a_{mj}} \lor λ_{a_{kn}})} \end{matrix}$

It is possible to directly obtain

$\tilde{X} = (\begin{matrix} (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) \end{matrix})$

Therefore, $\tilde{X}$ is the maximum element of A {1}.

In 2007, Khan and Pal proposed a method to find the g-inverse of IFMs in particular cases. In the next section, a new method is introduced to find all g-inverses with the help of the largest g-inverse in the general case.

4 The set of g-inverses

Note that a g-inverse of an IFM is not unique. The procedure to find all g-inverses of an IFM is described below.

Let A be a regular element of F_mn and $\begin{matrix} H_{il} = {h_{il} = (j_{il}, k_{il}) | μ_{a_{{ij}_{il}}} \land μ_{a_{k_{il} l}} \geq μ_{a_{il}}, μ_{{\tilde{x}}_{j_{il} k_{il}}} \\ \geq μ_{a_{il}}, λ_{a_{{ij}_{il}}} \lor λ_{a_{k_{il} l}} \leq λ_{a_{il}}, λ_{{\tilde{x}}_{j_{il} k_{il}}} \leq λ_{a_{il}}} \\ H = {H = (h_{il})_{mn} | h_{il} \in H_{il}} \end{matrix}$

For a $H \in H$ , let X_H ∈ F_nm, where $\begin{matrix} μ_{x_{H_{jk}}} & = \lor {μ_{a_{il}} | (j_{il}, k_{il}) = (j, k)}, λ_{x_{H_{jk}}} \\ = \land {λ_{a_{il}} | (j_{il}, k_{il}) = (j, k)}, \\ j = 1, 2, \dots, n, k = 1, 2, \dots, m \end{matrix}$

Theorem 4.1. Let A be a regular element of F_mn, then $H = {H = (h_{il})_{mn} | h_{il} \in H_{il}} \neq \emptyset$ , and the set of allg-inverses of A is $A {1} = {X | X_{H} \subseteq X \subseteq \tilde{X}, H \in H} .$

Proof. Let X be a generalized inverse (g-inverse) of A, then AXA = A and $X \subseteq \tilde{X}$ .

For all (i, l), $\begin{matrix} ⋁_{k} ⋁_{j} (μ_{a_{ij}} \land μ_{x_{jk}} \land μ_{a_{kl}}) \\ = μ_{a_{il}}, ⋀_{k} ⋀_{j} (λ_{a_{ij}} \lor λ_{x_{jk}} \lor λ_{a_{kl}}) = λ_{a_{il}} \end{matrix}$

There exists (j_il, k_il) such that $(μ_{a_{{ij}_{il}}} \land μ_{x_{j_{il} k_{il}}} \land μ_{a_{k_{il} l}}) = μ_{a_{il}}, (λ_{a_{{ij}_{il}}} \lor λ_{x_{j_{il} k_{il}}} \lor λ_{a_{k_{il} l}}) = λ_{a_{il}}$ Hence, there exists h_il = (j_il, k_il) such that $\begin{matrix} (μ_{a_{{ij}_{il}}} \land μ_{a_{k_{il} l}}) \geq μ_{a_{il}}, (λ_{a_{{ij}_{il}}} \lor λ_{a_{k_{il} l}}) \leq λ_{a_{il}}, μ_{{\tilde{x}}_{j_{il} k_{il}}} \\ \geq μ_{x_{j_{il} k_{il}}} \geq μ_{a_{il}}, λ_{{\tilde{x}}_{j_{il} k_{il}}} \leq λ_{x_{j_{il} k_{il}}} \leq λ_{a_{il}} \end{matrix}$

Clearly, $h_{il} \in H_{il}$ . There exists $H = (h_{il}) \in H,$ $H \neq \emptyset .$

Let $μ_{x_{H_{jk}}} = \lor {μ_{a_{il}} | (j_{il}, k_{il}) = (j, k)},$ $\begin{matrix} λ_{x_{H_{jk}}} = \land {λ_{a_{il}} | (j_{il}, k_{il}) = (j, k)}, \\ j = 1, 2, \dots, n, k = 1, 2, \dots, m \end{matrix}$

For any (j, k), if there exist some (j_il, k_il) = (j, k), then $\begin{matrix} μ_{x_{jk}} & = μ_{x_{j_{il} k_{il}}} \geq μ_{a_{il}}, λ x_{jk} = λ_{x_{j_{il} k_{il}}} \leq λ_{a_{il}} \\ μ_{x_{H_{jk}}} & = \lor {μ_{a_{il}} | (j_{il}, k_{il}) = (j, k)} \leq μ_{x_{jk}}, λ_{x_{H_{jk}}} \\ = \land {λ_{a_{il}} | (j_{il}, k_{il}) = (j, k)} \geq λ_{x_{jk}} \end{matrix}$

For all (j_il, k_il), if (j_il, k_il) ≠ (j, k), then $μ_{x_{H_{jk}}} = \lor \emptyset = 0 \leq μ_{x_{jk}}, λ_{x_{H_{jk}}} = \land \emptyset = 1 \geq λ_{x_{jk}}$

We have $X_{G} \subseteq X \subseteq \tilde{X}, X \in A {1}$ Conversely, for any $H = (h_{il}) \in H$ , $μ_{a_{i j^{'}}} \land μ_{a_{k^{'} l}} = μ_{a_{i j_{i l}}} \land μ_{a_{k_{i l} l}} \geq μ_{a_{i l}}$ $λ_{a_{i j^{'}}} \lor λ_{a_{k^{'} l}} = λ_{a_{i j_{i l}}} \lor λ_{a_{k_{i l} l}} \leq λ_{a_{i l}}$ $μ_{{\tilde{x}}_{j_{il} k_{il}}} \geq μ_{a_{il}}, λ_{{\tilde{x}}_{j_{il} k_{il}}} \leq λ_{a_{il}}$ for all (i, j) , h_il = (j_il, kil)=(j^′,k′) .

Then $\begin{array}{l} μ_{x_{H_{j^{'} k^{'}}}} = \lor {μ_{a_{s t}} | (j_{s t}, k_{s t}) = (j_{i l}, k_{i l})} \geq μ_{a_{i l}}, λ_{x_{H_{j^{'} k^{'}}}} \\ = \land {λ_{a_{s t}} | (j_{s t}, k_{s t}) = (j_{i l}, k_{i l})} \leq λ_{a_{i l}} \\ μ_{a_{i j^{'}}} \land μ_{x_{H_{j^{'} k^{'}}}} \land μ_{a_{k^{'} l}} \geq μ_{a_{i l}}, λ_{a_{i j^{'}}} \lor λ_{x_{H_{j^{'} k^{'}}}} \lor λ_{a i l} \leq λ_{a_{i l}} \\ \underset{k}{�} \underset{j}{�} (μ_{a_{i j}} \land μ_{x_{H_{j k}}} \land μ_{a_{k l}}) \geq μ_{a_{i j^{'}}} \land μ_{x_{H_{j^{'} k H'}}} \land μ_{a_{k^{'} l}} \geq μ_{a_{i l}} \\ \underset{k}{�} \underset{j}{�} (λ_{a_{i j}} \lor λ_{x_{H_{j k}}} \lor λ_{a_{k l}}) \leq λ_{a i j^{'}} \lor λ_{x_{H_{j^{'} k^{'}}}} \lor λ_{a_{k^{'} l}} \leq λ_{a_{i l}} \end{array}$ Finally, $A \subseteq A X_{H} A \subseteq A \tilde{X} A = A$

That is to say, for all X ∈ F_nm, which satisfies $X_{H} \subseteq X \subseteq \tilde{X},$ $A X_{H} A = AXA = A \tilde{X} A = A □$ Example 4.2. An intuitionistic fuzzy matrix $A = (\begin{matrix} (\begin{matrix} 0.5, & 0.2 \end{matrix}) (\begin{matrix} 0.5, & 0.2 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.4, & 0.3 \end{matrix}) (\begin{matrix} 0.4, & 0.3 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.5, & 0.2 \end{matrix}) \end{matrix})$

Problem-solving steps:

1 . Work out the largest g-inverse.

2 . In Table 1, the μ_{a
_il} are arranged in the order of small to large. In Table 2, the λ_{a
_il} are arranged in the order of large to small.

3 . In matrix A, if μ_{a
_ij} ∧ μ_{a
_kl} ≥ μ_{a
_il} and λ_{a
_ij} ∨ λ_{a
_kl} ≤ λ_{a
_il}, then fill μ_{a
_il} (λ_{a
_il}) in the cross section of the line of μ_{a
_il} (λ_{a
_il}) and column of x_jk in Table 1 (Table 2). Otherwise, leave the position open.

4 . For the filled μ_{a
_il} and λ_{a
_il}, if μ_{a
_il} ≥ μ_{x
_{H
_jk}} or λ_{a
_il} ≤ λ_{x
_{H
_jk}} in the same column, then delete them in table 1 and table 2, respectively. The two tables show h₃₁ = (j₃₁, k₃₁) ∈ {(1, 1) , (1, 2) , (1, 3) , (2, 1) , (2, 2) , (2, 3) , (3, 1) , (3, 2) , (3, 3)}, h₁₁ = (j₁₁, k₁₁) ∈ {(1, 1) (2, 1)} , ⋯⋯ (j₃₁, k₃₁) , (j₃₂, k₃₂) , (j₁₃, k₁₃) , (j₂₃, k₂₃) , (j₂₁, k₂₁) , (j₂₂, k₂₂) , (j₁₁, k₁₁) and (j₁₂, k₁₂)may be (1, 1), (j₃₁, k₃₁) , (j₃₂, k₃₂) , (j₁₃, k₁₃) , (j₂₃, k₂₃) , (j₂₁, k₂₁) and (j₂₂, k₂₂) may be (1, 2)⋯⋯

5 . In table 1, take an element of each line from the bottom to the top. If there are elements that have been chosen in the ith column, then the elements of the ith column have a priority to be selected. Take the supremum in each column. In the column whose elements are selected, if (j_il, k_il) = (j, k), then fill the supremum in the bottom. Otherwise, fill 0 in the bottom.

In table 2, take an element of each line from the bottom to the top. If there are elements that have been chosen in the ith column, then the elements of the ith column have a priority to be selected. Take the infimum in each column. In the column whose elements are selected, if (j_il, k_il) = (j, k), then fill the infimum in the bottom. Otherwise, fill 1 in the bottom. Then, it is possible to obtain X_H and all g-inverses of an IFM.

By the example 3.2, the largest g-inverse is $\tilde{X} = (\begin{matrix} (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) \end{matrix})$

By the Theorem 4.1, all the g-inverses of A can be obtained

$X_{g_{1}} = (\begin{matrix} (\begin{matrix} , & [0, 1] \end{matrix}) (\begin{matrix} 0, 1], & [0, 1] \end{matrix}) (\begin{matrix} 0, 0.3], & [0.4, 1] \end{matrix}) \\ (\begin{matrix} [0.5, 1], & [0, 0.2] \end{matrix}) (\begin{matrix} [0, 1], & [0, 1] \end{matrix}) (\begin{matrix} [0, 0.3], & [0.4, 1] \end{matrix}) \\ (\begin{matrix} [0, 0.2] & [0.5, 1] \end{matrix}) (\begin{matrix} [0, 0.2], & [0.5, 1] \end{matrix}) (\begin{matrix} [0.5, 1] & [0, 0.2] \end{matrix}) \end{matrix})$ $X_{g_{2}} = (\begin{matrix} (\begin{matrix} [0.5, 1], & [0, 0.2] \end{matrix}) (\begin{matrix} [0, 1], & [0, 1] \end{matrix}) (\begin{matrix} [0, 0.3], & [0.4, 1] \end{matrix}) \\ (\begin{matrix} [0, 1], & [0, 1] \end{matrix}) (\begin{matrix} [0, 1], & [0, 1] \end{matrix}) (\begin{matrix} [0, 0.3], & [0.4, 1] \end{matrix}) \\ (\begin{matrix} [0, 0.2] & [0.5, 1] \end{matrix}) (\begin{matrix} [0, 0.2], & [0.5, 1] \end{matrix}) (\begin{matrix} [0.5, 1] & [0, 0.2] \end{matrix}) \end{matrix})$

Example 4.3. An intuitionistic fuzzy matrix $A = (\begin{matrix} (\begin{matrix} 0.7, & 0.1 \end{matrix}) (\begin{matrix} 0.1, & 0.7 \end{matrix}) (\begin{matrix} 0.3, & 0.4 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.5, & 0.2 \end{matrix}) (\begin{matrix} 0.4, & 0.3 \end{matrix}) \end{matrix})$

$\tilde{X} = (\begin{matrix} (\begin{matrix} 1, & 0 \end{matrix}) (\begin{matrix} 0.1, & 0.7 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 1, & 0 \end{matrix}) \\ (\begin{matrix} 0.2, & 0.5 \end{matrix}) (\begin{matrix} 0.1, & 0.7 \end{matrix}) \end{matrix})$

All the g-inverses of A are $X_{g} = (\begin{matrix} (\begin{matrix} [0.7, 1], & [0, 0.1] \end{matrix}) (\begin{matrix} [0, 0.1], & [0.7, 1] \end{matrix}) \\ (\begin{matrix} [0, 0.2], & [0.5, 1] \end{matrix}) (\begin{matrix} [0.5, 1], & [0, 0.2] \end{matrix}) \\ (\begin{matrix} [0, 0.2], & [n 0.5, 1] \end{matrix}) (\begin{matrix} [0, 0.1], & [0.7, 1] \end{matrix}) \end{matrix})$

5 Conclusion

As a generalization of the notion of fuzzy matrices, the intuitionistic fuzzy matrix (IFM) is a new and important research object. The purpose of this paper is to deal with the problem of finding all the generalized inverses of a given IFM. In addition, a necessary and sufficient condition for the regularity of such a matrix is presented. It is hoped that this work will pave the way for further research on IFMs.

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