Fuzzy matrices with fuzzy rows and columns

Abstract

Fuzzy matrices (FMs) play important role to model several uncertain systems. In FMs, it is assumed that the rows and columns are certain. But, in many real life situations it is seen that they are also uncertain. So to model these types of uncertain problems, new type of FMs are introduced called fuzzy matrices with fuzzy rows and columns (FMFRCs). For these matrices, null, identity, equality, etc. are defined along with four binary operators ∨, ∧, ⊖ and ⊕. Two types of complements and density of FMFRC are defined and investigated several important properties. An application of FMFRC in image representation is also given.

Keywords

Fuzzy matrix fuzzy rows and columns complement of fuzzy matrices density of fuzzy matrices

1 Introduction

Like classical (crisp) matrices, fuzzy matrices (FMs) are now a very rich topic (it has a separate MSC 15B15) in modelling uncertain situations occurred in science, automata theory, logic of binary relations, medical diagnosis, etc. In FMs, only the elements are uncertain, while rows and columns are taken as certain. But, in many real life situations we observed that rows and columns may also be uncertain. For example, in a fuzzy graph the vertices and edges both are uncertain. If we represent a fuzzy graph in matrix form where the membership values of vertices and edges represents the membership values of rows and columns and elements represent the membership values of the corresponding edge. That is, in these matrices rows, columns and elements all are uncertain. We call these types of matrices are fuzzy matrices with fuzzy rows and columns (FMFRCs). This is the very new concept in fuzzy matrix theory.

FMs defined first time by Thomson in 1977 [39] and discussed about the convergence of the powers of a fuzzy matrix. The theories of fuzzy matrices were developed by Kim and Roush [23] as an extension of Boolean matrices. With max-min operation the fuzzy algebra and its matrix theory are considered by many authors [5 , 35]. Hashimoto [15] studied the canonical form of a transitive fuzzy matrix. Xin [40] studied controllable fuzzy matrices. Hemashina et al. [19] investigated iterates of fuzzy circulant matrices. Determinant theory, powers and nilpotent conditions of matrices over a distributive lattice are considered by Zhang [41] and Tan [38]. The transitivity of matrices over path algebra (i.e. additively idempotent semiring) is discussed by Hashimoto [16 –18]. Generalized fuzzy matrices, matrices over an incline and transitive closer, determinant, adjoint matrices, convergence of powers and conditions for nilpotency are considered by Duan [13] and Lur et al. [24]. Dehghan et al. [12] give two ideas for finding the inverse of a fuzzy matrix viz. scenario-based and arithmetic-based.

There are some limitations in dealing with uncertainties by fuzzy set. To overcome these difficulties, Atanassov [4] introduced theory of intuitionistic fuzzy set in 1993 as a generalization of fuzzy set. Based on this concept Pal et al. have defined intuitionistic fuzzy determinant in 2001 [27] and intuitionistic fuzzy matrices (IFMs) in 2002 [28]. Bhowmik and Pal [5] introduced some results on IFMs, intuitionistic circulant fuzzy matrix and generalized intuitionistic fuzzy matrix [5 –11]. Shyamal and Pal [34, 36] defined the distances between IFMs and hence defined a metric on IFMs. They also cited few applications of IFMs. In [26], the similarity relations, invertibility conditions and eigenvalues of IFMs are studied. Idempotent, regularity, permutation matrix and spectral radius of IFMs are also discussed. The parameterizations tool of IFM enhances the flexibility of its applications. For other works on IFMs see [1–3 , 36].

The concept of interval-valued fuzzy matrices (IVFMs) as a generalization of fuzzy matrix was introduced and developed in 2006 by Shaymal and Pal [37] by extending the max-min operation in fuzzy algebra. For more works on IVFMs see [30].

Combining IFMs and IVFMs, a new fuzzy matrix called interval-valued intuitionistic fuzzy matrices (IVIFMs) is defined [20]. For other works on IVIFMs, see [9, 11].

2 New definition of fuzzy matrices

In this section, a new concept of fuzzy matrix (FM) is introduced. In FM, the rows and columns are taken as crisp, i.e. it is assumed that there is no uncertainty on the rows and columns. But, in our new concept it is assumed that the rows and columns are uncertain, i.e. they have some membership values.

Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n be a fuzzy matrix with fuzzy rows and columns (FMFRC) of orderm × n. Here a_ij, i = 1, 2, …, m ; j = 1, 2, …, n represents the ijth elements of A, r_A (i) and c_A (j) represent the membership values of ith row and jth column respectively for i = 1, 2, …, m ; j = 1, 2, …, n.

In conventional way, a FMFRC can be written as

$\begin{array}{l} c_{A} (1) c_{A} (2) \dots c_{A} (n) \\ A = \begin{matrix} r_{A} (1) \\ r_{A} (2) \\ ⋮ \\ r_{A} (m) \end{matrix} [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ \dots & \dots & \dots & \dots \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}] \end{array}$

The membership values of the rows of a FMFRC A may be written as a fuzzy vector r_A = [r_A (1) , r_A (2) , …, r_A (m)] and similarly c_A = [c_A (1) , c_A (2) , …, c_A (n)].

This concept is also introduced for interval-valued fuzzy matrices [29].

If the value of r_A (i) or c_A (j) is 0 for some i or j, then it implies that the ith row or jth column has no importance for the FMFRC A and hence they may be removed from A. When r_A (i) =1 and c_A (j) =1 for all i, j, then FMFRC A becomes a FM.

2.1 Equality of FMFRC

The equality of two FMFRCs are defined in three different ways. Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)] [c_B (j)] [b_ij] _m×n be two FMFRCs.

Type I : If r_A (i) = r_B (i) and c_A (j) = c_B (j) for i = 1, 2, …, m ; j = 1, 2, …, n, whatever may be the relation between a_ij and b_ij, then we say that A and B are RC-equal and it is denoted by A = _RCB. If r_A (i) ≠ r_B (i) or c_A (j) ≠ c_B (j) for at least one i or j, then we say that A ≠ B in RC-equal sense. This is the weak equality between two FMFRCs and it occurs only in the FMFRCs.

Type II : If a_ij = b_ij for all i and j, whatever may be the values of r_A (i) , r_B (i) , c_A (j) , c_B (j), then A and B are e-equal and it is denoted by A = _eB. This type of equality occurs in FMs also.

If a_ij ≠ b_ij for at least one i or j, then we say A ≠ _eB or A ≠ B in e-equal sense.

Type III : If both A = _RCB and A = _eB, then we say that A and B are equal and it is denoted as A = B. That is, if

a_ij = b_ij for all i, j

r_A (i) = r_B (i) for all i

c_A (j) = c_B (j) for all j.

Then A is equal to B and is denoted as A = B. If A and B are not equal then it is denoted by A ≠ B. That is, if A ≠ _RCB and/or A ≠ _eB then we write A ≠ B.

2.2 Null FMFRC

Based on the membership values of rows, columns and elements, three types of null FMFRC are defined.

Type I : If r_A (i) =0, c_A (j) =0 and a_ij = 0 for all i and j, then FMFRC A is called p-null, denoted by 0_p. For example

$\begin{array}{l} 0 0 0 \\ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \end{array}$

is a 3 × 3 order p-null FMFRC. This concept is same as FMs as well as classical matrices.

Type II : If a_ij = 0 for all i and j, whatever may be the values of r_A (i) and c_A (j), then the FMFRC A is called e-null, and it is denoted by 0_e. For example

$\begin{array}{l} 0.6 0.1 0.8 \\ \begin{matrix} 0.5 \\ 0.4 \\ 0.2 \end{matrix} [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \end{array}$

is a 3 × 3 order e-null FMFRC.

This concept is also similar to FMs.

Type III : If r_A (i) =0, c_A (j) =0 for all i and j, whatever may be the values of a_ij, then the FMFRC is called RC-null and it is denoted by 0_RC.

For example

$\begin{array}{l} 0 0 0 0 \\ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} [\begin{matrix} 0.2 & 0.1 & 0.3 \\ 0.3 & 0.8 & 0.1 \\ 0.5 & 0.2 & 0.5 \end{matrix} \begin{matrix} 0.50 \\ 0.85 \\ 0.71 \end{matrix}] \end{array}$

is a RC-null FMFRC.

This type of null matrix is new and it is only defined for FMFRC.

2.3 Identity FMFRC

Two types of identity FMFRC are defined here.

Type I : A square FMFRC of order n × n is calledp-identity FMFRC if r_A (i) =1 and c_A (j) =1 for all i and j and a_ii = 1, a_ij = 0, i ≠ j for all i, j. It is denoted by I_p. For example

$\begin{array}{l} 1 1 1 \\ \begin{matrix} 1 \\ 1 \\ 1 \end{matrix} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{array}$

is a 3 × 3 order p-identity FMFRC.

Type II : A square FMFRC of order n × n is calledf-identity if a_ii = 1, a_ij = 0, i ≠ j for all i, j, whatever may be the values of r_A (i) and c_A (j) and it is denoted by I_f. For example

$\begin{array}{l} 0.2 0.3 0.4 \\ \begin{matrix} 0.8 \\ 0.7 \\ 0.1 \end{matrix} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{array}$

is a 3 × 3 order f-identity FMFRC.

3 Operators on FMFRCs

Throughout the paper the symbols ∨ and ∧ represent the maximum and minimum between two elements, i.e. $a \lor b = \max {a, b} and a \land b = \min {a, b} .$

3.1 ∨ operator

Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)][c_B (j)] [b_ij] _m×n be two FMFRCs. Then $A \lor B = D = [r_{D} (i)] [c_{D} (j)] [d_{ij}]_{m \times n},$ where r_D (i) = r_A (i) ∨ r_B (i), c_D (j) = c_A (j) ∨ c_B (j) and d_ij = a_ij ∨ b_ij for all i, j.

Note that the order of A and B must be equal.

But, in our new concept one can operate two FMFRCs with different orders. Suppose A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)] [c_B (j)] [b_ij] _p×q. For the sack of simplicity, we assume that m ≤ p and n ≤ q. If m = p and n = q then there is nothing new.

Otherwise, three different cases may arise:

m < p, n ≤ q

m ≤ p, n < q

m < p, n < q.

In these cases, add p - m (may be 0) rows and q - n (may also be 0) columns at the end of rows and columns.

The elements of these rows and columns are taken as 0 and membership values of all rows and columns are taken as 0. After introduction of these rows and columns, the FMFRC A becomes a FMFRC of order p × q. Now, the ∨ operation can be performed as in previous case. Note that the order of A ∨ B becomes p × q.

To illustrate this new concept, let us consider two FMFRCs as

$\begin{array}{l} \begin{matrix} 0.1 & 0.7 & 0.8 \end{matrix} \\ A = \begin{matrix} 0.2 \\ 0.3 \end{matrix} [\begin{matrix} 0.1 & 0.8 \\ 0.2 & 0.9 \end{matrix} \begin{matrix} 0.7 \\ 0.3 \end{matrix}] \end{array}$ and $\begin{array}{l} \begin{matrix} 0.5 & 0.6 \end{matrix} \\ B = \begin{matrix} 0.2 \\ 0.8 \\ 0.3 \end{matrix} [\begin{matrix} 0.1 \\ 0.8 \\ 0.3 \end{matrix} \begin{matrix} 0.2 \\ 0.6 \\ 0.7 \end{matrix}] \end{array}$

Here the number of rows of A is less than that of B and the number of columns of B is less than that of A. The augmented matrices A_a and B_a are given by

A_a = $\begin{array}{l} 0.1 0.7 0.8 \\ \begin{matrix} 0.2 \\ 0.3 \\ 0.0 \end{matrix} [\begin{matrix} 0.1 & 0.8 & 0.7 \\ 0.2 & 0.9 & 0.3 \\ 0.0 & 0.0 & 0.0 \end{matrix}] \end{array}$

and

B_a = $\begin{array}{l} 0.5 0.6 0.0 \\ \begin{matrix} 0.2 \\ 0.8 \\ 0.3 \end{matrix} [\begin{matrix} 0.1 & 0.2 & 0.0 \\ 0.8 & 0.6 & 0.0 \\ 0.3 & 0.7 & 0.0 \end{matrix}] \end{array}$

Now,

A ∨ B = $\begin{array}{l} 0.5 0.7 0.8 \\ \begin{matrix} 0.2 \\ 0.8 \\ 0.3 \end{matrix} [\begin{matrix} 0.1 & 0.8 & 0.7 \\ 0.8 & 0.9 & 0.3 \\ 0.3 & 0.7 & 0.0 \end{matrix}] \end{array}$

Observe that the order of A and B are 2 × 3 and 3 × 2, but the order of A ∨ B is 3 × 3.

Observations: It is easy to verify the following results:

0_p ∨ A = A

0_e ∨ A ≠ A

0_e ∨ A ≠ _RCA

0_e ∨ A = _eA

0_RC ∨ A ≠ A

0_RC ∨ A ≠ _RCA

0_RC ∨ A = _eA.

3.2 ∧ operator

The ∧ operation is similar to ∨ operation.

Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)][c_B (j)] [b_ij] _m×n be two FMFRCs. Then $A \land B = D = [r_{D} (i)] [c_{D} (j)] [d_{ij}]_{m \times n},$ where

r_D (i) = r_A (i) ∧ r_B (i),

c_D (j) = c_A (j) ∧ c_B (j) and

d_ij = a_ij ∧ b_ij for all i, j.

If the orders of A and B are different, then this case can be handled as in case of ∨ operation.

3.3 ⊖ operator

Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)][c_B (j)] [b_ij] _m×n be two FMFRCs. Then A ⊖ B =D = [r_D (i)] [c_D (j)] [d_ij] _m×n, where

$r_{D} (i) = {\begin{matrix} r_{A} (i) & if r_{A} (i) > r_{B} (i) \\ 0 & otherwise \end{matrix}$

$c_{D} (j) = {\begin{matrix} c_{A} (j) & if c_{A} (j) > c_{B} (j) \\ 0 & otherwise \end{matrix}$ and $d_{ij} = {\begin{matrix} a_{ij} & if a_{ij} > b_{ij} \\ 0 & otherwise . \end{matrix}$

Example 1. Let

A = $\begin{array}{l} \begin{matrix} 0.3 & 0.8 \end{matrix} \\ \begin{matrix} 0.5 \\ 0.6 \end{matrix} [\begin{matrix} 0.9 & 0.6 \\ 0.2 & 0.5 \end{matrix}] \end{array}$

and

B = $\begin{array}{l} \begin{matrix} 0.7 & 0.5 \end{matrix} \\ \begin{matrix} 0.5 \\ 0.3 \end{matrix} [\begin{matrix} 0.8 & 0.6 \\ 0.3 & 0.4 \end{matrix}] \end{array}$

Then

A ⊖ B = $\begin{array}{l} \begin{matrix} 0.0 & 0.8 \end{matrix} \\ \begin{matrix} 0.0 \\ 0.6 \end{matrix} [\begin{matrix} 0.9 & 0.0 \\ 0.0 & 0.5 \end{matrix}] \end{array}$

4 g-FMFRC

In this section, a special type of FMFRC is defined along with other two types of FMFRCs.

Definition 1. If a_ij ≤ r_A (i) ∧ c_A (j) for all i, j, then the FMFRC A = [r_A (i)] [c_A (j)] [a_ij] _m×n is calledg-FMFRC.

For example, the FMFRCs

A = $\begin{array}{l} 0.2 0.5 0.8 \\ \begin{matrix} 0.2 \\ 0.5 \\ 0.8 \end{matrix} [\begin{matrix} 0.2 & 0.1 & 0.2 \\ 0.1 & 0.4 & 0.3 \\ 0.2 & 0.3 & 0.7 \end{matrix}] \end{array}$

and

B = $\begin{array}{l} 0.50 0.60 0.90 \\ \begin{matrix} 0.3 \\ 0.8 \\ 0.7 \end{matrix} [\begin{matrix} 0.25 & 0.42 & 0.30 \\ 0.61 & 0.60 & 0.80 \\ 0.60 & 0.62 & 0.68 \end{matrix}] \end{array}$

both are g-FMFRC.

Definition 2. If a_ij = r_A (i) ∧ c_A (j) for all i, j, then the FMFRC A is called complete FMFRC.

From definition it is obvious that every complete FMFRC is g-FMFRC, but converse is not true. The FMFRC

$\begin{array}{l} \begin{matrix} 0.5 & 0.2 & 0.7 \end{matrix} \\ A = \begin{matrix} 0.3 \\ 0.8 \end{matrix} [\begin{matrix} 0.3 & 0.2 & 0.3 \\ 0.5 & 0.2 & 0.7 \end{matrix}] \end{array}$

is a complete FMFRC.

Now we define another kind of FMFRC.

Definition 3. If a_ij ≤ r_A (i) • c_A (j) for all i, j, then the FMFRC A is called dot FMFRC, where •denotes ordinary multiplication.

The FMFRC

A = $\begin{array}{l} 0.80 0.70 0.30 0.60 \\ \begin{matrix} 0.5 \\ 0.3 \\ 0.4 \end{matrix} [\begin{matrix} 0.40 & 0.30 & 0.10 \\ 0.20 & 0.10 & 0.00 \\ 0.32 & 0.20 & 0.10 \end{matrix} \begin{matrix} 0.30 \\ 0.11 \\ 0.22 \end{matrix}] \end{array}$

is a dot FMFRC.

Lemma 1. Every dot FMFRC is a g-FMFRC.

Proof. Since 0 ≤ r_A (i) ≤1 and 0 ≤ c_A (i) ≤1, r_A (i) • c_A (j) ≤ r_A (i) ∧ c_A (j) for all i, j.

If A is a dot FMFRC, then a_ij ≤ r_A (i) • c_A (j) for all i, j. Therefore, for all i, j, a_ij ≤ r_A (i) • c_A (j) ≤ r_A (i) ∧ c_A (j), i.e. a_ij ≤ r_A (i) ∧ c_A (j) for all i, j.

Hence, A is a g-FMFRC. □

5 Complement of FMFRC

In this section, two types of complements of a FMFRC are defined. Like fuzzy matrix, the complement of a FMFRC is defined below:

Definition 4. Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n be a FMFRC. Its complement is denoted by A^c and it is defined as A^c = [1 - r_A (i)] [1 - c_A (j)] [1 - a_ij] _m×n, where ‘-’ represents the ordinary subtraction. We call this complement as c-complement.

For a g-FMFRC, the complement can be defined in another way.

Definition 5. Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n be a g-FMFRC. The complement of the g-FMFRC A is denoted by $\bar{A}$ and it is defined as $\bar{A} = [\bar{r_{A} (i)}] [\bar{c_{A} (j)}] [\bar{a_{ij}}]_{m \times n}$ , where $\bar{r_{A} (i)} = r_{A} (i), \bar{c_{A} (j)} = c_{A} (j)$ and $\bar{a_{ij}} = r_{A} (i) \land c_{A} (j) - a_{ij}$ for all i, j, here also ‘-’ represents the ordinary subtraction.

This complement is defined only for g-FMFRC and we call it b-complement.

From definition of $\bar{A}$ it is easy to verify that $\bar{A} =_{RC} A$ for all g-FMFRCs.

Again, if A is a complete g-FMFRC, then $\bar{A} = [r_{A} (i)] [c_{A} (j)] [0]$ , as for complete g-FMFRC, a_ij = r_A (i) ∧ c_A (j) for all i, j. Thus, for a complete g-FMFRC, $\bar{A} =_{RC} A$ and $\bar{A}$ is e-null FMFRC.

Theorem 1. If A is a g-FMFRC, then $\bar{\bar{A}} = A$ .

Proof. Since $\bar{r_{A} (i)} = r_{A} (i)$ and $\bar{c_{A} (j)} = c_{A} (j)$ , therefore $\bar{\bar{r_{A} (i)}} = \bar{r_{A} (i)} = r_{A} (i)$ and $\bar{\bar{c_{A} (j)}} = \bar{c_{A} (j)} = c_{A} (j)$ .

Again, $\bar{\bar{a_{ij}}} = \bar{r_{A} (i)} \land \bar{c_{A} (j)} - \bar{a_{ij}} = r_{A} (i) \land c_{A} (j) - (r_{A} (i) \land c_{A} (j) - a_{ij}) = a_{ij} .$

Hence, $\bar{\bar{A}} = A$ . □

For some FMFRCs it may happen that $\bar{A}$ and A are same.

Definition 6. A g-FMFRC A is called self b-complement if $\bar{A} = A$ .

Remember that b-complement is defined only for g-FMFRC. Therefore, self b-complement is also defined only for g-FMFRC.

Theorem 2. If A is a self b-complement FMFRC, then $a_{ij} = \frac{1}{2} r_{A} (i) \land c_{A} (j)$ for all i, j.

Proof. Since $\bar{A} = A$ , therefore $\bar{r_{A} (i)} = r_{A} (i)$ , $\bar{c_{A} (j)} = c_{A} (j)$ and $\bar{a_{ij}} = a_{ij}$ for all i, j.

This gives, r_A (i) ∧ c_A (j) - a_ij = a_ij for all i, j.

That is, $a_{ij} = \frac{1}{2} r_{A} (i) \land c_{A} (j)$ for all i, j. □

From this theorem, the following result is obvious.

Corollary 1. If A is a self b-complement FMFRC, then $\sum_{i \neq j} a_{ij} = \frac{1}{2} \sum_{i \neq j} r_{A} (i) \land c_{A} (j)$ .

But, the converse of this result is not true.

This is justified in the following example.

Let

A = $\begin{array}{l} \begin{matrix} 0.5 & 1 & 1 \end{matrix} \\ \begin{matrix} 0.5 \\ 1 \\ 1 \end{matrix} [\begin{matrix} 0 & 0.5 & 0.25 \\ 0.5 & 0 & 0.25 \\ 0.25 & 0.25 & 0 \end{matrix}] \end{array}$ ,

$\bar{A} =$ $\begin{array}{l} \begin{matrix} 0.5 & 1 & 1 \end{matrix} \\ \begin{matrix} 0.5 \\ 1 \\ 1 \end{matrix} [\begin{matrix} 0.5 & 0 & 0.25 \\ 0 & 1 & 0.75 \\ 0.25 & 0.75 & 1 \end{matrix}] \end{array}$

In this case, $\sum_{i \neq j} a_{ij} = 2 and \frac{1}{2} \sum_{i \neq j} r_{A} (i) \land c_{A} (j) = 2 .$

That is, $\sum_{i \neq j} a_{ij} = \frac{1}{2} \sum_{i \neq j} r_{A} (i) \land c_{A} (j), but \bar{A} \neq A .$

Like self b-complement, the self c-complement can also be defined.

Definition 7. A FMFRC is called self c-complement if A^c = A.

Theorem 3. If A is a FMFRC, then (A^c) ^c = A.

Proof. Let B = A^c. Then r_B (i) =1 - r_A (i) , c_B (j) =1 - c_A (j) , b_ij = 1 - a_ij. If D = B^c = (A^c) ^c, then r_D (i) =1 - r_B (i) = r_A (i) , c_D (j) = 1 - c_B (j) = c_A (j) and d_ij = 1 - b_ij = a_ij.

Hence, D = A, i.e. (A^c) ^c = A. □

Theorem 3. If a FMFRC A = [r_A (i)] [c_A (j)] [a_ij] _m×n is self c-complement, then $r_{A} (i) = \frac{1}{2}, c_{A} (j) = \frac{1}{2}$ and $a_{ij} = \frac{1}{2}$ for all i, j.

Proof. By the definition of c-complement, $\bar{r_{A} (i)} = 1 - r_{A} (i)$ , $\bar{c_{A} (j)} = 1 - c_{A} (j)$ , $\bar{a_{ij}} = 1 - a_{ij}$ for all i, j.

Since, A^c = A, r_A (i) =1 - r_A (i) or, $r_{A} (i) = \frac{1}{2}$ .

Similarly, $c_{A} (j) = \frac{1}{2}$ and $a_{ij} = \frac{1}{2}$ for all i, j. □

Theorem 5. (De Morgan’s laws)

Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)] [c_B (j)] [b_ij] _m×n be two FMFRCs. Then

(A ∨ B) ^c = A^c ∧ B^c

(A ∧ B) ^c = A^c ∨ B^c.

Proof. (i) Let D = A ∨ B. Then r_D (i) = r_A (i) ∨ r_B (i) , c_D (j) = c_A (j) ∨ c_B (j) , d_ij = a_ij ∨ b_ij.

Let E = D^c. Then r_E (i) =1 - r_D (i) =1 - r_A (i) ∨ r_B (i), c_E (j) =1 - c_D (j) =1 - c_A (j) ∨ c_B (j),

e_ij = 1 - d_ij = 1 - a_ij ∨ b_ij.

Let F = A^c ∧ B^c.

Therefore, r_F (i) = {1 - r_A (i)} ∧ {1 - r_B (i)} =1 - r_A (i) ∨ r_B (i) = r_E (i), c_F (j) = {1 - c_A (j)} ∧ {1 - c_B (j)} =1 - c_A (j) ∨ c_B (j) = c_E (j) and f_ij =(1 - a_ij) ∧ (1 - b_ij) =1 - a_ij ∨ b_ij = e_ij.

Hence, (A ∨ B) ^c = A^c ∧ B^c.

(ii) Proof is similar to (i). □

It can be verified that the above results are not true for b-complement.

Example 2. Let

$\begin{array}{l} \begin{matrix} 0.8 & 0.2 & 0.6 \end{matrix} \\ A = \begin{matrix} 0.2 \\ 0.6 \end{matrix} [\begin{matrix} 0.1 & 0.2 \\ 0.5 & 0.2 \end{matrix} \begin{matrix} 0.1 \\ 0.4 \end{matrix}] \end{array}$

and

$\begin{array}{l} \begin{matrix} 0.3 & 0.4 & 0.5 \end{matrix} \\ B = \begin{matrix} 0.8 \\ 0.5 \end{matrix} [\begin{matrix} 0.4 & 0.5 \\ 0.4 & 0.2 \end{matrix} \begin{matrix} 0.5 \\ 0.6 \end{matrix}] \end{array}$

Then,

A ∨ B = $\begin{array}{l} \begin{matrix} 0.8 & 0.4 & 0.5 \end{matrix} \\ \begin{matrix} 0.8 \\ 0.6 \end{matrix} [\begin{matrix} 0.4 & 0.5 \\ 0.4 & 0.5 \end{matrix} \begin{matrix} 0.5 \\ 0.6 \end{matrix}] \end{array}$

(A ∧ B) ^c = $\begin{array}{l} \begin{matrix} 0.2 & 0.6 & 0.5 \end{matrix} \\ \begin{matrix} 0.2 \\ 0.4 \end{matrix} [\begin{matrix} 0.6 & 0.5 \\ 0.6 & 0.5 \end{matrix} \begin{matrix} 0.5 \\ 0.4 \end{matrix}] \end{array}$

(A) ^c = $\begin{array}{l} \begin{matrix} 0.2 & 0.8 & 0.5 \end{matrix} \\ \begin{matrix} 0.8 \\ 0.4 \end{matrix} [\begin{matrix} 0.9 & 0.8 \\ 0.5 & 0.8 \end{matrix} \begin{matrix} 0.9 \\ 0.6 \end{matrix}] \end{array}$

(B) ^c = $\begin{array}{l} \begin{matrix} 0.7 & 0.6 & 0.5 \end{matrix} \\ \begin{matrix} 0.2 \\ 0.5 \end{matrix} [\begin{matrix} 0.6 & 0.5 \\ 0.6 & 0.8 \end{matrix} \begin{matrix} 0.5 \\ 0.6 \end{matrix}] \end{array}$

A^c ∧ B^c = $\begin{array}{l} \begin{matrix} 0.2 & 0.6 & 0.5 \end{matrix} \\ \begin{matrix} 0.2 \\ 0.4 \end{matrix} [\begin{matrix} 0.6 & 0.5 \\ 0.6 & 0.5 \end{matrix} \begin{matrix} 0.5 \\ 0.4 \end{matrix}] \end{array}$

Thus, (A ∨ B) ^c = A^c ∧ B^c.

Let us define a new operator ⊕ for the FMFRCs A = [r_A (i)] [c_A (j)] [a_ij] _m×n and B = [r_B (i)] [c_B (j)] [b_ij] _m×n as

A ⊕ B = (A^c ∧ B) ∨ (A ∧ B^c) .

Theorem 6. For any FMFRCs A and B, A^c ⊕ B^c = A ⊕ B.

Proof. By definition A^c ⊕ B^c = {(A^c) ^c ∧ B^c} ∨ {A^c ∧ (B^c) ^c}

= (A ∧ B^c) ∨ (A^c ∧ B) = A ⊕ B. □

6 Density of FMFRC

Sometimes we see that a matrix, it may be classical matrix or fuzzy matrix, contains many zeros. This type of matrix is called sparse matrix and special methods are used to store such matrix in computer. In contrast, if the number of non-zero elements is high then the matrix is called dense matrix.

The density of a FM is define below.

Definition 8. Let A = [a_ij] _m×n be a FM. The density of A is denoted as D (A) and is defined as $D (A) = \frac{1}{mn} \sum_{i, j} a_{ij} .$

From definition it follows that 0 ≤ D (A) ≤1 for a FM A. Actually, D (A) represents the average membership of the elements in the FM A.

But, in FMFRC, rows and columns are not certain, and hence the density is to be re-defined for FMFRC. The definition is given below.

Definition 9. Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n be a FMFRC. The density of A is denoted as D (A) and is defined as $D (A) = \frac{\sum_{i, j} a_{ij}}{\sum_{i, j} r_{A} (i) \land c_{A} (j)},$ provided ∑_i,jr_A (i) ∧ c_A (j) ≠0.

For a FMFRC, a_ij ≥ 0 for all i, j. Thus, D (A) is non-negative for any FMFRC A. D (A) is zero only when all a_ij = 0. Higher value of D (A), indicates the matrix is more dense. If the value of D (A) is closed to zero, then the FMFRC is called sparse FMFRC with degree of sparsity D (A).

Unlike FM, the value of D (A) for FMFRC is not necessity less than or equal to 1. For example, for the FMFRC

A = $\begin{array}{l} \begin{matrix} 0.2 & 0.3 \end{matrix} \\ \begin{matrix} 0.5 \\ 0.6 \end{matrix} [\begin{matrix} 0.6 & 0.7 \\ 0.3 & 0.3 \end{matrix}] \end{array}$ $D (A) = \frac{0.6 + 0.7 + 0.3 + 0.3}{0.2 + 0.3 + 0.2 + 0.3} = 1.9 > 1 .$

But, for a g-FMFRC, the density is bounded and its upper bound is 1.

Theorem 7. Let A be a g-FMFRC. Then 0 ≤ D (A) ≤1.

Proof. For any g-FMFRC, a_ij ≤ r_A (i) ∧ c_A (j) for all i, j.

Therefore, ∑_i,ja_ij ≤ ∑_i,jr_A (i) ∧ c_A (j) i.e. $\frac{\sum_{i, j} a_{ij}}{\sum_{i, j} r_{A} (i) \land c_{A} (j)} \leq 1 .$

Hence, D (A) ≤1.

Again, a_ij ≥ 0 and ∑_i,jr_A (i) ∧ c_A (j) ≠0, so, D (A) ≥0.

Hence, for any FMFRC A, 0 ≤ D (A) ≤1. □

Theorem 8. If A is a complete g-FMFRC, thenD (A) =1 .

Proof. For any complete g-FMFRC, a_ij = r_A (i) ∧ c_A (j) for all i, j. Therefore, $\sum_{i, j} a_{ij} = \sum_{i, j} r_{A} (i) \land c_{A} (j) .$

Hence, D (A) =1. □

Like classical sub-matrix one can defined sub-FMFRC. Any portion, not necessarily consecutive, along with corresponding rows and columns membership values is called sub-FMFRC.

Definition 10. A FMFRC A is called balanced if D (S) ≤ D (A) for all sub-FMFRCs S of A.

Let

A = $\begin{array}{l} \begin{matrix} 0.8 & 0.5 \end{matrix} \\ \begin{matrix} 0.6 \\ 0.7 \end{matrix} [\begin{matrix} x & 0.5 \\ 0.7 & 0.5 \end{matrix}] \end{array}$

Let

S₁ = $\begin{array}{l} 0.8 \\ 0.6 [x] \end{array}$ S₂ = $\begin{array}{l} 0.5 \\ 0.6 [0.5] \end{array}$

S₃ = $\begin{array}{l} 0.8 \\ 0.7 [0.7] \end{array}$ ,

S₄ = $\begin{array}{l} 0.5 \\ 0.7 [0.5] \end{array}$ ,

S₅ = $\begin{array}{l} \begin{matrix} 0.8 & 0.5 \end{matrix} \\ \begin{matrix} 0.6 \\ 0.7 \end{matrix} [\begin{matrix} x & 0.5 \\ 0.7 & 0.5 \end{matrix}] \end{array}$

be the sub-FMFRCs of A.

Now, $D (S_{1}) = \frac{x}{0.6}$ , D (S₂) = 1, D (S₃) = 1, D (S₄) =1, $D (S_{5}) = \frac{x + 1.7}{2.3}$ .

If x = 0.3, then D (S₁) =0.5 and $D (S_{5}) = \frac{2}{2.3}$ . In this case, A is not balanced FMFRC. But, if x = 0.6, then D (S₁) =1 and D (S₅) =1 and hence A is balanced FMFRC.

Now, we define a particular type of balanced FMFRC.

Definition 11. A FMFRC A is called strictly balanced if D (S) = D (A) for all sub-FMFRCs S of A.

Example 3.Let

A = $\begin{array}{l} \begin{matrix} 0.3 & 0.2 \end{matrix} \\ \begin{matrix} 0.3 \\ 0.2 \end{matrix} [\begin{matrix} 0.225 & 0.150 \\ 0.150 & 0.150 \end{matrix}] \end{array}$ .

Then D (A) =0.75.

Let S₁ = $\begin{array}{l} 0.3 \\ 0.3 [0.225] \end{array}$ , S₂ = $\begin{array}{l} 0.3 \\ 0.2 [0.15] \end{array}$ ,S₃ = $\begin{array}{l} 0.2 \\ 0.3 [0.15] \end{array}$ , S₄ = $\begin{array}{l} 0.2 \\ 0.2 [0.15] \end{array}$ .

Therefore, D (S₁) =0.75, D (S₂) =0.75, D (S₃) =0.75, D (S₄) =0.75.

Thus, D (S_i) = D (A) =0.75 for all i. Hence, A is strictly balanced FMFRC.

Theorem 9. Every complete g-FMFRC is strictly balanced.

Proof. Let A be a complete g-FMFRC. Therefore, a_ij = r_A (i) ∧ c_A (j) for all i, j.

Therefore, ∑_i,ja_ij = ∑_i,jr_A (i) ∧ c_A (j).

Thus, D (A) =1.

That is, for any complete g-FMFRC A, D (A) =1.

Again, it is obvious that every sub-FMFRC of a complete g-FMFRC is complete. Thus for any sub-FMFRC S of A, D (S) =1.

Hence, A is strictly balanced. □

Let

A = $\begin{array}{l} \begin{matrix} 0.5 & 0.6 \end{matrix} \\ \begin{matrix} 0.7 \\ 0.4 \end{matrix} [\begin{matrix} 0.5 & 0.6 \\ 0.4 & 0.4 \end{matrix}] \end{array}$

be a complete FMFRC.

Then D (A) =1. Also,

S₁ = $\begin{array}{l} 0.5 \\ 0.7 [0.5] \end{array}$ , S₂ = $\begin{array}{l} 0.6 \\ 0.7 [0.6] \end{array}$ ,

S₃ = $\begin{array}{l} 0.5 \\ 0.4 [0.4] \end{array}$ , S₄ = $\begin{array}{l} 0.6 \\ 0.4 [0.4] \end{array}$ .

Therefore, D (S₁) =1, D (S₂) =1, D (S₃) =1, D (S₄) =1.

Thus, D (S_i) = D (A) =1 for all i. Hence, A is strictly balanced FMFRC.

But, the converse of this theorem is not true, i.e. every balanced FMFRC is not necessarily complete. This is justified in Example 3.

In the following, we prove a very interesting property about the density of g-FMFRC.

Theorem 10. Let A be a g-FMFRC and $\bar{A}$ be itsb-complement. Then $D (A) + D (\bar{A}) = 1$ .

Proof. Let A be a g-FMFRC. Since, $\bar{A}$ is the b-complement of the g-FMFRC, therefore $\bar{a_{ij}} = r_{A} (i) \land c_{A} (j) - a_{ij}, \bar{r_{A} (i)} = r_{A} (i), \bar{c_{A} (j)} = c_{A} (j)$ for all i, j.

Now, $D (\bar{A}) = \frac{\sum_{i, j} \bar{a_{ij}}}{\sum_{i, j} \bar{r_{A} (i)} \land \bar{c_{A} (j)}}$

$= \frac{\sum_{i, j} [r_{A} (i) \land c_{A} (j) - a_{ij}]}{\sum_{i, j} r_{A} (i) \land c_{A} (j)}$

$= 1 - \frac{\sum_{i, j} a_{ij}}{\sum_{i, j} r_{A} (i) \land c_{A} (j)} = 1 - D (A) .$

Hence, $D (A) + D (\bar{A}) = 1$ . □

Theorem 11. The b-complement of a strictly balanced FMFRC is strictly balanced.

Proof. Let A be a strictly balanced FMFRC and S be any sub-FMFRC of A. Then by definition D (A) = D (S). Let $\bar{S}$ be any sub-FMFRC of the b-complement $\bar{A}$ of A. The ijth element $\bar{s_{ij}}$ of $\bar{S}$ is given by $\bar{s_{ij}} = r_{S} (i) \land c_{S} (j) - s_{ij}$ .

Therefore,

$D (\bar{S}) = \frac{\sum_{i, j} {r_{S} (i) \land c_{S} (j) - s_{ij}}}{\sum_{i, j} r_{S} (i) \land c_{S} (j)}$

$= 1 - \frac{\sum_{i, j} s_{ij}}{\sum_{i, j} r_{S} (i) \land c_{S} (j)}$

= 1 - D (S) =1 - D (A) as D (A) = D (S).

Again, $D (\bar{A}) = 1 - D (A)$ by Theorem 10.

Thus, $D (\bar{A}) = D (\bar{S})$ for any sub-FMFRC $\bar{S}$ of $\bar{A}$ .

Hence, $\bar{A}$ is strictly balanced. □

Theorem 12. Let A = [r_A (i)] [c_A (j)] [a_ij] _m×n be a FMFRC and A^c be its c-complement. Then D (A) + D (A^c) ≥1.

Proof. By definition $D (A) = \frac{\sum_{i, j} a_{ij}}{\sum_{i, j} r_{A} (i) \land c_{A} (j)}$ and $D (A^{c}) = \frac{\sum_{i, j} (1 - a_{ij})}{\sum_{i, j} (1 - r_{A} (i)) \land (1 - c_{A} (j))}$ .

Now, r_A (i) ∧ c_A (j) ≤1 for a fixed i and j.

Therefore, ∑_i,jr_A (i) ∧ c_A (j) ≤ mn. Similarly, ∑_i,j (1 - r_A (i)) ∧ (1 - c_A (j)) ≤ mn.

Thus,

$D (A) + D (A^{c}) \geq \frac{\sum_{i, j} a_{ij}}{mn} + \frac{\sum_{i, j} (1 - a_{ij})}{mn}$

$= \frac{\sum_{i, j} {a_{ij} + (1 - a_{ij})}}{mn} = 1$ . □

7 Application of FMFRC

In this section, an application of FMFRC is described. FMFRC can be successfully used to represent an image. Also, it is used to image contraction. Any plane image can be treated as a matrix. The grey value of a pixel is consider as an element of the corresponding matrix. The grey value depends on the colour of a pixel and it can be represented within the unit interval [0, 1]. Thus, the value of each element of the matrix lies on [0, 1]. Sometimes, it is difficult to measure the grey value of a pixel due to hesiness of the image or defect of the instrument or improper snap, etc. Thus, the grey value is a fuzzy number. Again, certain portion of the entire image may not be significant, i.e. that portion contains only background colour. While the other portion may be highly significant and some another portion is less significant, etc. Depending on this analogy one can graded each row and each column of the matrix corresponding to the given image. Thus, the rows and columns are also uncertain and hence they have some membership values. This type of image can be represented as a FMFRC. To demonstrate this fact, we consider the image of Fig. 1. The FMFRC for this image is shown in Table 1 (due to lack of space, first 64 columns are given). The complement of the FMFRC of Table 1 is shown in Table 2 (first 16 columns are given) and the corresponding image is shown in Fig. 2. This image is the complement of the image of Fig. 1. If the membership value of a row or column is zero, then that row and column may be removed to reduce the size of the matrix/image.

8 Conclusion

In this article, very new kind of fuzzy matrices has been introduced. In this new approach, the rows and columns are taken as uncertain, whereas in fuzzy matrices these are certain. These types of fuzzy matrices can be used to handel images, fuzzy graphs, etc.

In this article, null-FMFRC, equality of FMFRCs and identity FMFRC are defined. Four operators, viz. ∧, ∨, ⊖ and ⊕ are defined. Two types of complements and density of FMFRC are defined and studied several properties. But, product of two FMFRCs is not defined. At present, we are working on different types of product of FMFRCs and their properties. The power convergence, nilpotency, etc. can be investigated after suitable definition of multiplication. Other several operations can also be defined for FMFRC.

References

Adak

A.K.

, Bhowmik

and Pal

, Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattice, Fuzzy Inf and Eng 4(4) (2012), 371–387.

Adak

A.K.

, Bhowmik

and Pal

, Intuitionistic fuzzy block matrix and its some properties, Annals of Pure and Applied Mathematics 1(1) (2012), 13–31.

Adak

A.K.

, Pal

and Bhowmik

, Distributive lattice over intuitionistic fuzzy matrices, The Journal of Fuzzy Mathematics 21(2) (2013), 401–416.

Atanassov

K.T.

, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, Bulgarian, 1983.

Bhowmik

and Pal

, Generalized intuitionistic fuzzy matrices, Far-East Journal of Mathematical Sciences 29(3) (2008), 533–554.

Bhowmik

and Pal

, Some results on intuitionistic fuzzy matrices and circulant intuitionistic fuzzy matrices, International Journal of Mathematical Sciences 7(1–2) (2008), 81–96.

Bhowmik

, Pal

and Pal

, Circulant triangular fuzzy number matrices, Journal of Physical Sciences 12 (2008), 141–154.

Bhowmik

and Pal

, Intuitionistic neutrosophic set, Journal of Information and Computing Science 4(2) (2009), 142–152.

Bhowmik

and Pal

, Intuitionistic neutrosophic set relations and some of its properties, Journal of Information and Computing Science 5(3) (2010), 183–192.

10.

Bhowmik

and Pal

, Generalized interval-valued intuitionistic fuzzy sets, The Journal of Fuzzy Mathematics 18(2) (2010), 357–371.

11.

Bhowmik

and Pal

, Some results on generalized intervalvalued intuitionistic fuzzy sets, International Journal of Fuzzy Systems 14(2) (2012), 193–203.

12.

Dehghan

, Ghatee

and Hashemi

, The inverse of a fuzzzy matrix of fuzzy numbers, International Journal of Computer Mathematics 86(8) (2009), 1433–1452.

13.

Duan

J-S

, The transitive closure, convergence of powers and adjoint of generalized fuzzy matrices, Fuzzy Sets and Systems 145 (2004), 301–311.

14.

Gavalec

, Computing matrix period in max-min algebra, Discrete Appl Math 75 (1997), 63–70.

15.

Hashimoto

, Canonical form of a transitive fuzzy matrix, Fuzzy Sets and Systems 11 (1983), 157–162.

16.

Hashimoto

, Convergence of power of a fuzzy transitive matrix, Fuzzy Sets and Systems 9 (1983), 153–160.

17.

Hashimoto

, Canonical form of a transitive fuzzy matrix, Fuzzy Sets and Systems 11 (1983), 157–162.

18.

Hashimoto

, Transitivity of generalized fuzzy matrix, Fuzzy Sets and Systems 17 (1985), 83–90.

19.

Hemasinha

and Pal

N.R.

, Bezdek

J.C.

, Iterates of fuzzy circulant martices, Fuzzy Sets and Systems 60 (1993), 199–206.

20.

Khan

S.K.

and Pal

, Interval-valued intuitionistic fuzzy matrices, Notes on Intuitionistic Fuzzy Sets 11(1) (2005), 16–27.

21.

Khan

S.K.

and Pal

, Some operations on intuitionistic fuzzy matrices, Acta Ciencia Indica XXXII M(2) (2006), 515–524.

22.

Khan

S.K.

and Pal

, The generalized inverse of intuitionistic fuzzy matrix, Journal of Physical Sciences 11 (2007), 62–67.

23.

Kim

K.H.

and Roush

F.W.

, Generalized fuzzy matrices, Fuzzy Sets and Systems 4 (1980), 293–315.

24.

Lur

Y-Y

, Pang

C-T

and Guu

S-M

, On nilpotent fuzzy matrices, Fuzzy Sets and Systems 145 (2004), 287–299.

25.

Mondal

and Pal

, Intuitionistic fuzzy incline matrix and determinant, Annals of Fuzzy Mathematics and Informatics 8(1) (2014), 19–32.

26.

Mondal

and Pal

, Similarity relations, invertibility and eigenvalues of intuitoinistic fuzzy matrix, Fuzzy Inf Eng 4 (2013), 431–443.

27.

Pal

, Intuitionistic fuzzy determinant, VUJ Physical Sciences 7 (2001), 87–93.

28.

Pal

, Khan

S.K.

and Shyamal

A.K.

, Intuitionistic fuzzy matrices, Notes on Intuitionistic Fuzzy Sets 8(2) (2002), 51–62.

29.

Pal

, Interval-valued fuzzy matrices with interval-valued fuzzy rows and columns, to appear in Fuzzy Engineering and Information.

30.

Pal

and Pal

, Some results on interval-valued fuzzy matrices, The 2010 International Conference on E-Business Intelligence, Org by Tsinghua University, Kunming, China, Atlantis Press 2010, pp. 554–559. doi:10.2991/icebi.2010.39

31.

Pradhan

and Pal

, Intuitionistic fuzzy linear transformations, Annals of Pure and Applied Mathematics 1(1) (2012), 57–68.

32.

Pradhan

and Pal

, Generalized inverse of block intuitionistic fuzzy matrices, International Journal of Applications of Fuzzy Sets and Artificial Intelligence 3 (2013), 23–38.

33.

Pradhan

and Pal

, Convergence of maxgeneralized meanmingeneralized mean powers of intuitionistic fuzzy matrices, The Journal of Fuzzy Mathematics 22(2) (2013), 477–492.

34.

Shyamal

A.K.

and Pal

, Distances between intuitionistic fuzzy matrices, VUJ Physical Sciences 8 (2002), 81–91.

35.

Shyamal

A.K.

and Pal

, Two new operations on fuzzy matrices, Journal of Applied Mathematics and Computing 15(1-2) (2004), 91–107.

36.

Shyamal

A.K.

and Pal

, Distance between intuitionistic fuzzy matrices and its applications, Natural and Physical Sciences 19(1) (2005), 39–58.

37.

Shyamal

A.K.

and Pal

, Interval-valued fuzzy matrices, The Journal of Fuzzy Mathematics 14(3) (2006), 583–604.

38.

Tan

Y-J

, Eigenvalues and eigenvectors for matrices over distributive lattice, Linear Algebra Appl 283 (1998), 257–272.

39.

Thomson

M.G.

, Convergence of powers of a fuzzy matrix, Journal of Mathematical Analysis and Applications 57 (1977), 476–480.

40.

Xin

L.J.

, Convergence of power of a controllble fuzzy matrices, Fuzzy Sets and System 45 (1992), 313–319.

41.

Zhang

K.L.

, Determinant theory for D01-lattice matrices, Fuzzy Sets and Systems 62 (1994), 347–353.