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Intuitionistic fuzzy relations compatible with the group Zn

Abstract

In this paper, we define the compatibility of finite intuitionistic fuzzy relations with the group Zn and prove some of their fundamental properties. We show that some compositions of Zn-compatible intuitionistic fuzzy relations are also Zn-compatible intuitionistic fuzzy relation. Also, from any given finite intuitionistic fuzzy relation ρ, we can construct two intuitionistic fuzzy relations denoted by ρL and ρU which are compatible with Zn. We have also provided some examples to clarify the notions and results.

Introduction

In 1965, Zadeh [1] came out with the concept of fuzzy subsets (or fuzzy sets for briefly) which is, indeed, an extension of the classical notion of the ordinary sets. This concept was defined by Zadeh as a function A : X → [0, 1] where X is a non-empty set. Here, A is a fuzzy subset of X and the number A(x)  [0, 1] is called the degree of membership of the element x in A for every xX. Zadeh also in [2] defined the fuzzy relation as a fuzzy subset of the Cartesian product X × Y. Subsequently, many researchers [3,4,5,6,7] and others studied fuzzy relations in various contexts. Fuzzy sets and fuzzy relations have many applications in diverse types of sciences, for example, in data bases, pattern recognition, neural network, fuzzy modelling, economy, medicine, and multicriteria decision-making. The concept of Zadeh has, however, some limitations in dealing with uncertainties. Atanasov [8] developed the theory of intuitionistic fuzzy sets as a generalization of fuzzy sets introduced by Zadeh to overcome these difficulties. Atanasov defined the intuitionistic fuzzy set as a pair of fuzzy sets, namely a membership and non-membership functions, which represent positive and negative aspects of the given information. After that, many researchers applied the notion of intuitionistic fuzzy sets to topology, algebra, and other branches of mathematics. Intuitionistic fuzzy set theory is widely applied in solving real-life problems. An example of such application is the optimization in intuitionistic fuzzy environment where by applying this concept, it is possible to reformulate the optimization problem by using degrees of rejection of constraints and values of the objective which are non-admissible. This concept allows one to define the degree of rejection which cannot be simply a complement of the degree of acceptance. The idea of a positive and negative information was confirmed by psychological investigations and is widely studied in diverse domains of engineering.

In this paper, we introduce the concept of compatibility of intuitionistic fuzzy relation with the well-known group (Zn, ) where is the sum modulo n. However, we obtain a characterizations of intuitionistic fuzzy relation compatible with the group (Zn,  ) throughout the isomorphic group (In, ) where In = {1, 2, …, n}.

The paper is organized in four sections. The first section is devoted to the introduction of the paper and the second section is devoted to introduce some definitions and some operations on the ordinary fuzzy relations and intuitionistic fuzzy relations. In the third section, which is the main in this paper, some important results are obtained using the operations and notations introduced in Section 2. The last section is devoted to the conclusion.

Basic Definitions

We give here some definitions and notations which are applied throughout the paper.

Definition 2.1. [1] Let X be a non-empty set. A function A from X to the unit interval [0, 1] is called a fuzzy subset of.X. The number A(x) is called a membership grade of x in A for every xX

Definition 2.2. [2] Let X be a non-empty set. A function R from X × X to the unit interval [0, 1] is called a fuzzy relation on X.

If card X = n where n, then R may be represented by a matrix [0, 1]n × n (called a fuzzy matrix). There is a large number of papers on fuzzy matrices from 1977 to now.

Definition 2.3. [8] Let X be a non-empty set and A, Ad : X → [0, 1] be two fuzzy subsets of X such that A(x) + Ad(x) ≤ 1 for every xX. A pair \( \mathcal{F}=\left\langle A,{A}^d\right\rangle \) is called an intuitionistic fuzzy subset of X where A(x) denotes the degree of membership and Ad(x) denotes the degree of non-membership of every xX to \( \mathcal{F} \) respectively.

Definition 2.4. [4] Let X be a nonempty set and R, Rd : X × X → [0, 1] be two fuzzy relations on X such that R(x, y) + Rd(x, y) ≤ 1 for every x, yX. A pair ρ = 〈R, Rdis called an intuitionistic fuzzy relation on X.

Similarly, as of ordinary fuzzy relations, if card X = n where n, then ρ may be represented by a matrix ([0, 1] × [0, 1])n × n (called an intuitionistic fuzzy matrix). Note that the elements of such matrices are taken from the set

$$ F=\left\{\left\langle a,{a}^d\right\rangle :a+{a}^d\le 1\ \right\}\ \mathrm{such}\ \mathrm{that}\ a,{a}^d\in \left[0,1\right] $$

The family of all intuitionistic fuzzy relations on a non-empty set X is denoted by IFR(X). If ρ = 〈R, Rd〉 is an intuitionistic fuzzy relation with R(x, y) + Rd(x, y) ≤ 1 for every x, yX, then ρ is reduced to be ordinary fuzzy relation. This explains why the concept of intuitionistic fuzzy relations is an extension of that of the ordinary fuzzy ones. If the fuzzy relation πρ : X × X → [0, 1] is associated to each intuitionistic fuzzy relation ρ = 〈R, Rd〉, where πρ(x, y) = 1 − R(x, y) − Rd(x, y) for every x, yX, the number πρ(x, y) is called an index of an element (x, y) in the intuitionistic fuzzy relation ρ. It is also described as an index (a degree) of hesitation whether x and y are in the relation ρ or not. This value is also regarded as a measure of non-determinacy or uncertainty (see [11]), and it is useful in some applications namely in decision-making problems.

From Definition 2. 4, we see that an intuitionistic fuzzy relation is a pair of fuzzy relations which represent a membership and non-membership functions, respectively. This description is connected with the existence of positive and negative information about the given relation. The symbols R(x, y) and Rd(x, y) may be regarded as a lower bound on membership and a lower bound on non-membership of a pair (x, y), respectively.

Some basic operations on intuitionistic fuzzy relations IFR(X) are extensions of the respective operations on ordinary fuzzy relations. As a result, operations on the set of ordinary fuzzy relations are particular cases of the ones on the set IFR(X).

Definition 2.5. [3]. Let ρ = 〈R, Rd〉, σ = 〈S, Sd〉 IFR(X). Then, the following operations are defined as follows:

$$ \mathrm{sum}:\rho \vee \sigma =\left\langle R\vee S,{R}^d\wedge {S}^d\right\rangle, $$
$$ \mathrm{intersection}:\rho \wedge \sigma =\left\langle R\wedge S,{R}^d\vee {S}^d\right\rangle, $$
$$ \mathrm{composition}:\rho \circ \sigma =\left\langle R\circ S,{R}^d{\circ}^{\prime }{S}^d\right\rangle, $$
$$ \mathrm{complement}:{\rho}^c=\left\langle {R}^d,R\right\rangle $$

where

$$ \left(R\vee S\right)\left(x,y\right)=R\left(x,y\right)\vee S\left(x,y\right)=\max \left(R\left(x,y\right),S\left(x,y\right)\right)x,y\in X $$
$$ \left(R\wedge S\right)\left(x,y\right)=R\left(x,y\right)\wedge S\left(x,y\right)=\min \Big(R\left(x,y\right),S\left(x,y\right)\kern0.5em x,y\in X $$
$$ \left(R\circ S\right)\left(x,y\right)={\mathrm{V}}_{z\in X}\Big(R\left(x,z\right)\wedge S\left(z,y\right) $$
$$ \left({R}^d{\circ}^{\prime }{S}^d\right)\left(x,y\right)=\underset{z\in X}{\wedge}\Big({R}^d\left(x,z\right)\vee {S}^d\left(z,y\right) $$

We may write ρ2 = ρρ, ρ3 = ρ2ρ, and in general, ρk = ρk − 1ρ for every k. If ρ, σIFR(X), then we write ρ ≤ σ if and only if ρ(x, y) ≤ σ(x, y). That is ρ ≤ σ if and only if R(x, y) ≤ S(x, y) and Rd(x, y) ≥ Sd(x, y) for all x, yX and ρ = σ if and only if R(x, y) = S(x, y) and Rd(x, y) = Sd(x, y) for all x, yX. Also, ρ < σ if and only if R(x, y) < S(x, y) and Rd(x, y) > Sd(x, y) for all x, yX

Definition 2.6. [4, 12] Let ρ = 〈R, Rd〉 IFR(X). Then, the converse intuitionistic fuzzy relation of ρ is denoted by ρ−1 and is defined as ρ−1 = 〈R−1, (Rd)−1, where R−1(x, y) = R(y, x) and (Rd)−1(x, y) = Rd(y, x) for every x, yX.

Definition 2.7. [4, 12] Let ρ = 〈R, Rd〉 IFR(X).

Then ρ is called an:

(a) intuitionistic fuzzy symmetric if and only if ρ−1=ρ,

(b) intuitionistic fuzzy transitive if and only if ρ2 ≤ ρ, i.e., ρ(x, y) ρ(y, z) ≤ ρ(x, z) for all x, y, zX,

(c) intuitionistic fuzzy reflexive if and only if ρ(x, x) = 〈1, 0〉 for all xX,

(d) intuitionistic fuzzy irreflexive if and only if ρ(x, x) = 〈0, 1〉 for all xX.

Definition 2.8. [7, 9, 10] Let G be a multiplicative semi-group. A fuzzy relation R on G is called a left G-compatible fuzzy relation if R(xa, xb) ≥ R(a, b) for every a, b, xG, and it is called a right G-compatible fuzzy relation if R(ax, bx) ≥ R(a, b) for every a, b, xG. It is called a G-compatible fuzzy relation if it is left and right G-compatible fuzzy relation.

Zn-Compatible Intuitionistic Fuzzy Relations

Let \( {Z}_n=\left\{{l}_1=\overline{0},{l}_2=\overline{1},\dots, {l}_n=\overline{n-1}\right\} \). Then, as it is well known, (Zn, ) is an abelian group where the operation is the sum modulo n. Let us put lilj = lij for every i, jIn = {1, 2, …, n} where n. Then, (In, ) is also an abelian group where the operation is defined on the set In as follows:

$$ i\ast j=\left\{\begin{array}{c}i+j-1\kern0.5em if\ i+j-1\le n,\\ {}i+j-1-n\kern0.5em if\ i+j-1>n.\end{array}\right. $$

The mapping φ : (Zn, ) → (In, ) defined by φ(lilj) = ij is an isomorphism of groups and the two groups are isomorphic, and so, the intuitionistic fuzzy relations defined on the group (Zn, ) may be regarded as that of the isomorphic group (In, ). In the following, we write Zn instead of (Zn, ) for simplicity.

Definition 3.1.

The relation ρ = 〈R, Rd〉 IFR(In, ) is called a left Zn-compatible if and only if R(i, j) ≤ R(ki, kj) and Rd(i, j) ≥ Rd(ki, kj) for every i, j, kIn. Similarly, ρ is called a right Zn-compatible if and only if R(i, j) ≤ R(ik, jk) and Rd(i, j) ≥ Rd(ik, jk) for every i, j, kIn. From this definition, since the operation is abelian, we see that when the intuitionistic fuzzy relation ρ is a left Zn-compatible, it is also a right Zn--compatible and then we call it Zn-compatible. Moreover, since, (In, ) is a group and ρ is a Zn-compatible, we get

$$ R\left(i,j\right)\le R\left(i\ast k,j\ast k\right)=R\left(i\ast k\ast {k}^{-1},j\ast k\ast {k}^{-1}\right)=R\left(i,j\right) $$

and

$$ {R}^d\left(i,j\right)\ge {R}^d\left(i\ast k,j\ast k\right)={R}^d\left(i\ast k\ast {k}^{-1},j\ast k\ast {k}^{-1}\right)={R}^d\left(i,j\right) $$

Thus, an intuitionistic fuzzy relation ρ = 〈R, Rd〉 is called a Zn-compatible if and only if R(i, j) = R(ik, jk) and Rd(i, j) = Rd(ik, jk) for every i, j, kIn, so that R(i, i) = R(j, j) and Rd(i, i) = Rd(j, j) for every i, j, kIn. i.e., ρ(i, i) = ρ(j, j) for every i, j, kIn. Here, the two fuzzy relations R and Rd are also Zn-compatible.

Remark 3.1. When the two fuzzy relations R and Rd in Definition 3.1 are presented by fuzzy matrices, then these fuzzy matrices are completely determined by their first rows (or first column) and so we may write bi to express ρ(1, i) = 〈R(1, i), Rd(1, i)〉.

Proposition 3.1. Let ρ = 〈R, Rd〉 IFR(In, ). Then, ρ is a Zn-compatible if and only if ρ(i, j) = ρ(ik, jk) for every i, j, kIn.

Proof. First, we notice that ik = ik 2 for every i, j, kIn. Now, let ρ be an intuitionistic fuzzy relation compatible with the group Zn and let i, j, kIn. Then, by the commutativity of and noting that k 2 In, we have:

$$ \rho \left(i,j\right)=\rho \Big(i\ast \left(k\ast 2\right),j\ast \left(k\ast 2\right)=\rho \left(i\bigoplus k,j\bigoplus k\right). $$

Conversely, suppose that ρ(i, j) = ρ(ik, jk) for every, j, kIn. Since knIn and k 2 = 1, we have:

$$ \rho \left(i,j\right)=\rho \left(i\bigoplus \left(k\ast n\right),j\bigoplus \left(k\ast n\right)\right) $$
$$ =\rho \Big(i\ast \left(k\ast n\right)\ast 2,j\ast \left(k\ast n\Big)\ast 2\right)=\rho \left(i\ast k,j\ast k\right). $$

Therefore, ρ is a Zn-compatible intuitionistic fuzzy relation. ☐

Proposition 3.2.

The following statements are equivalent:

(1) ρ is a Zn − compatible intuitionistic fuzzy relation,

(2) \( \rho \left(i,j\right)={b}_{i^{-1}\ast j} \),

(3) \( \mathrm{if}\ \rho \left(i,j\right)={b}_k,\mathrm{then}\ \rho \left(j,i\right)={b}_{k^{-1}}\ \mathrm{for}\ \mathrm{every}\ i,j,k\in {I}_n. \)

Proof. (1) implies (2).

Let ρ be a Zn-compatible intuitionistic fuzzy relation. Then

ρ(i, j) = ρ(ik, jk) for every i, j, kIn

and so

$$ \rho \left(i,j\right)=\rho \left(i\ast {i}^{-1},j\ast {j}^{-1}\right)=\rho \left(1,j\ast {i}^{-1}\right)={b}_{j\ast {i}^{-1}}={b}_{i^{-1}\ast j}. $$

(2) implies (1).

Let \( \rho \left(i,j\right)={b}_{i^{-1}\ast j} \) for every i, j, kIn.

Then,

$$ \kern3.25em \rho \left(i\ast k,j\ast k\right)={b}_{{\left(i\ast k\right)}^{-1}\ast \left(j\ast k\right)}={b}_{k^{-1}\ast {i}^{-1}\ast j\ast k}={b}_{i^{-1}\ast j}=\rho \left(i,j\right). $$

Thus, ρ is a Zn-compatible intuitionistic fuzzy relation.

(2) implies (3).

Let \( \rho \left(i,j\right)={b}_{i^{-1}\ast j} \) and let ρ(i, j) = bk for every i, j, kIn. Then,

$$ \rho \left(i,j\right)={b}_k={b}_{i^{-1}\ast j} $$

and so

$$ {b}_{k^{-1}}={b}_{{\left({i}^{-1}\ast j\right)}^{-1}}={b}_{j^{-1}\ast i}=\rho \left(j,i\right). $$

(3) implies (2).

Suppose that \( \rho \left(j,i\right)={b}_{k^{-1}} \) whenever ρ(i, j) = bk for every i, j, kIn. We will show that \( \rho \left(i,j\right)={b}_{i^{-1}\ast j} \).

Since \( {b}_{1^{-1}\ast i}={b}_i=\rho \left(1,i\right) \), we get

$$ \rho \left(i,1\right)={b}_{i^{-1}}={b}_{i^{-1}\ast 1}, $$

Thus,

\( \kern5.5em \rho \left(1,i\right)={b}_{1^{-1}\ast i} \) and \( \rho \left(i,1\right)={b}_{i^{-1}\ast 1} \).

Therefore,

\( \rho \left(i,j\right)={b}_{i^{-1}\ast j} \)

Remarks 3.2. (1 ) It follows from Proposition 3.2 that the number of elements of a Z n -compatible intuitionistic fuzzy relation is at most equal n.

(2) If a relation ρ is presented by an intuitionistic fuzzy matrix, then we can obtain this matrix from the table of (Zn, ) by replacing the ith row by i−1 th row and l by b.

(3) By Proposition 3.1, we can see that:

(i) ρ(i, j) = b(n + j − i + 1) with bn + l = bl ,

(ii) ρ(i, j) = ρ(ik, jk) = ρ(i + 1, j + 1) with ρ(n + l, n + h) = ρ(1, h) for every i, j, kIn,

(iii) ρ(n, j) = bj + 1) for every jIn.

(4) The rows parallel to the diagonal have the same elements.

Example 3.1. If =4 , then \( {Z}_4=\left\{{l}_1=\overline{0},{l}_2=\overline{1},{l}_3=\overline{2},{l}_4=\overline{3}\right\} \) and the table of (Z4, ) is as follows:

$$ {\displaystyle \begin{array}{ccccc}\oplus & {l}_1& {l}_2& {l}_3& {l}_4\\ {}{l}_1& {l}_1& {l}_2& {l}_3& {l}_4\\ {}{l}_2& {l}_2& {l}_3& {l}_4& {l}_1\\ {}{l}_3& {l}_3& {l}_4& {l}_1& {l}_2\\ {}{l}_4& {l}_4& {l}_1& {l}_2& {l}_3\end{array}} $$

Since \( {l}_1^{-1}={l}_{1,\kern0.5em }\ {l}_2^{-1}={l}_4,\kern0.5em and\ {l}_3^{-1}={l}_3 \), it follows from Proposition 3.1 that the following matrix:

$$ \rho =\left[\begin{array}{c}{b}_1\kern0.5em {b}_2\kern0.5em {b}_3\kern0.5em {b}_4\\ {}{b}_4\kern0.5em {b}_1\ {b}_2\kern0.50em {b}_3\\ {}{b}_3\kern0.5em {b}_{4\kern0.5em }{b}_1\kern0.5em {b}_2\\ {}{b}_2\kern0.5em {b}_{3\kern0.5em }{b}_4\kern0.5em {b}_1\end{array}\right] $$

represents an intuitionistic fuzzy relation compatible with the group Z4 where biF for all i = 1, 2, 3, 4. This matrix is called circulant as it is well known.

Proposition 3.3.

Let ρ = 〈R, Rdand σ = 〈S, Sdbe two Zn-compatible intuitionistic fuzzy relations. Then, ρ−1, ρc, ρσ, ρσ and ρσ are also intuitionistic fuzzy relations.

Proof. We only prove that the composition of ρ and σ is also a Zn-compatible intuitionistic fuzzy relation and the proof of the others are straightforward. Now,

$$ \rho \circ \sigma \left(i,j\right)=\left\langle {\bigvee}_{l=1}^n\Big(R\left(i,l\right)\wedge S\left(l,j\right),\underset{l=1}{\overset{n}{\bigwedge \limits }}\left({R}^d\left(i,l\right)\vee {S}^d\right(l,j\Big)\right\rangle $$

Since we have that R, S, Rd and Sd are all Zn-compatible fuzzy relations, we write

$$ \rho \circ \sigma \left(i,j\right)=\left\langle {\bigvee}_{l=1}^n\Big(R\left(i\ast k,l\ast k\right)\wedge S\left(l\ast k,j\ast k\right),\underset{l=1}{\overset{n}{\bigwedge \limits }}\left({R}^d\left(i\ast k,l\ast k\right)\vee {S}^d\right(l\ast k,j\ast k\Big)\right\rangle $$
$$ =\left\langle {\bigvee}_{h=1}^n\Big(R\left(i\ast k,h\right)\wedge S\left(h,j\ast k\right),\underset{h=1}{\overset{n}{\bigwedge \limits }}\left({R}^d\left(i\ast k,h\right)\vee {S}^d\right(h,j\ast k\Big)\right\rangle $$

(since (In, ) is a group)

$$ =\rho \circ \sigma \left(i\ast k,j\ast k\right). $$

Therefore, ρσ is a Zn-compatible intuitionistic fuzzy relation. ☐

Remarks 3.3. By Proposition 3.4., we have:

(1) If ρ is a Zn-compatible intuitionistic fuzzy relation, then so also ρk for all k.

(2) If β is the set of all Zn-compatible intuitionistic fuzzy relations, then (β, ), (β, ), and (β, ) are monoids with unite element ε1, ε2,and ε3 respectively, where ε1, ε2,and ε3 and are defined as ε1(i, j) = 〈1, o〉,  ε2(i, j) = 〈0, 1〉, and

$$ {\varepsilon}_3\left(i,j\right)=\left\{\begin{array}{c}\left\langle 1,0\right\rangle \kern0.5em if\ i=j,\\ {}\left\langle 0,1\right\rangle \kern0.5em if\ i\ne j\end{array}\right. $$

for every i, jIn.

Definition 3.2. Let ρ = 〈R, Rd〉, σ = 〈S, Sd〉 IFR(In). Then, we define ρσ, ρσIFR(In) as

$$ \left(\rho \ominus \sigma \right)\left(i,j\right)=\left\{\begin{array}{c}\left\langle 0,1\right\rangle \kern0.5em \mathrm{if}\ R\left(i,j\right)\le S\left(i,j\right)\kern0.5em \mathrm{and}\ {R}^d\left(i,j\right)\ge {S}^d\left(i,j\right),\kern2.75em \\ {}\left\langle 0,{R}^d\left(i,j\right)\right\rangle\ \mathrm{if}\ R\left(i,j\right)\le S\left(i,j\right)\ \mathrm{and}\ {R}^d\ \left(i,j\right)<{S}^d\left(i,j\right),\\ {}\left\langle R\left(i,j\right),{R}^d\left(i,j\right)\right\rangle\ if\ R\left(i,j\right)>S\left(i,j\right).\kern8.25em \end{array}\right. $$

and

$$ \left(\rho \vartriangleright \sigma \right)\left(i,j\right)=\underset{k=1}{\overset{n}{\bigwedge \limits }}\Big(\left\langle R\left(i,l\right),{R}^d\left(i,l\right)\right\rangle \vartriangleright \left\langle S\left(l,j\right),{S}^d\left(l,j\right)\right\rangle $$

where

$$ \left\langle R\left(i,j\right),{R}^d\left(i,j\right)\right\rangle \vartriangleright \left\langle S\left(l,k\right).{S}^d\left(l,k\right)\right\rangle $$

=\( \left\{\begin{array}{c}\left\langle 1,0\right\rangle \kern0.5em \mathrm{if}\ R\left(i,j\right)\le S\left(l,k\right),\kern16.25em \\ {}\left\langle S\left(l,k\right),0\right\rangle \kern0.5em \mathrm{if}\ R\left(i,j\right)>S\left(l,k\right)\ \mathrm{and}\ {R}^d\left(i,j\right)\ge {S}^d\left(l,k\right),\kern3em \\ {}\left\langle S\left(l,k\right),{S}^d\left(l,k\right)\ \right\rangle \kern0.5em \mathrm{if}\ \mathrm{R}\left(\mathrm{i},\mathrm{j}\right)>\mathrm{S}\left(\mathrm{l},\mathrm{k}\right)\ \mathrm{and}\ {R}^d\left(i,j\right)<{S}^d\left(l,k\right)\ \end{array}\right. \)

for every i, j, kIn

Proposition 3.4.

Let ρ = 〈R, Rd〉 and σ = 〈S, Sdbe two Zn-compatible intuitionistic fuzzy relations. Then, ρσ and ρσ are also Zn-compatible intuitionistic fuzzy relations.

Proof. To prove that ρσ is a Zn-compatible intuitionistic fuzzy relation, let τ = ρσ. Then,

$$ \tau \left(i,j\right)=\left\langle T\left(i,j\right),{T}^d\left(i,j\right)\right\rangle $$
$$ \kern0.75em =\left\{\begin{array}{c}\left\langle 0,1\right\rangle \kern0.5em \mathrm{if}\ \mathrm{R}\left(\mathrm{i},\mathrm{j}\right)\le \mathrm{S}\left(\mathrm{i},\mathrm{j}\right)\kern0.5em \mathrm{and}\ {R}^d\left(i,j\right)\ge {S}^d\left(i,j\right)\kern3.5em \\ {}\left\langle 0.{R}^d\left(i,j\right)\right\rangle \kern0.5em \mathrm{if}\ R\left(i,j\right)\le S\left(i,j\right)\kern0.5em \mathrm{and}\ {R}^d\left(i,j\right)<{S}^d\left(i,j\right),\\ {}\left\langle R\left(i,j\right),{R}^d\left(i,j\right)\ \right\rangle\ \mathrm{if}\ R\left(i,j\right)>S\left(i,j\right).\kern8.5em \end{array}\right. $$

Now, we have three cases:

Case (i). If R(i, j) ≤ S(i, j) and Rd(i, j) ≥ Sd(i, j), then

$$ R\left(i\ast k,j\ast k\right)\le S\left(i\ast k,j\ast k\right) $$

and

$$ {R}^d\ \left(i\ast k,j\ast k\right)\ge {S}^d\left(i\ast k,j\ast k\right). $$

Therefore,

τ(i, j) = 〈0, 1〉=τ(ik, jk).

Case (ii). If R(i, j) ≤ S(i, j) and Rd(i, j) < Sd(i, j), then

$$ R\left(i\ast k,j\ast k\right)\le S\left(i\ast k,j\ast k\right) $$

and

$$ {R}^d\ \left(i\ast k,j\ast k\right)<{S}^d\left(i\ast k,j\ast k\right). $$

Therefore,

$$ \tau \left(i,j\right)=\left\langle 0,{R}^d\left(i,j\right)\right\rangle =\left\langle 0,{R}^d\left(i\ast k,j\ast k\right)\right\rangle =\tau \left(i\ast k,j\ast k\right) $$

Case (iii). If R(i, j) > S(i, j), then R(ik, jk) > S(ik, jk) and so

$$ \tau \left(i\ast k,j\ast k\right)=\left\langle R\left(i\ast k,j\ast k\right),{R}^d\left(i\ast k,j\ast k\right)\right\rangle $$
$$ =\left\langle R\left(i,j\right),{R}^d\left(i,j\right)\right\rangle =\tau \left(i,j\right). $$

Therefore, ρσ is a Zn-compatible intuitionistic fuzzy relation.

To show that ρσ is a Zn-compatible intuitionistic fuzzy relation, let η = ρσ. Then, we have

$$ \eta \left(i,j\right)=\underset{l=1}{\overset{n}{\bigwedge \limits }}\left(\left\langle R\left(i,l\right),{R}^d\left(i,l\right)\right\rangle \vartriangleright \left\langle S\left(l,j\right),{S}^d\left(l,j\right)\right\rangle \right)\eta \left(i,j\right) $$
$$ =\underset{l=1}{\overset{n}{\bigwedge \limits }}\left(\left\langle R\left(i\ast k,l\ast k\right),{R}^d\left(i\ast k,l\ast k\right)\right\rangle \vartriangleright \left\langle S\left(l\ast k,j\ast k\right),{S}^d\left(l\ast k,j\ast k\right)\right\rangle \right) $$
$$ =\underset{h=1}{\overset{n}{\bigwedge \limits }}\left(\left\langle R\left(i\ast k,h\right),{R}^d\left(i\ast k,h\right)\right\rangle \vartriangleright \left\langle S\left(h,j\ast k\right),{S}^d\left(h,j\ast k\right)\right\rangle \right) $$

(since (In, ) is a group)

=η(ik, jk) for every i, j, kIn.ρ.

Thus, the intuitionistic fuzzy relation ρσ is a Zn-compatible. ☐

Corollary 3.1. Let ρ be a Zn-compatible intuitionistic fuzzy relation. Then, ρ and ∆ρ are Zn-intuitionistic fuzzy relations where ρ = ρρ−1 and ∆ρ = ρρ−1 .

Proof. By Propositions 3.3 and 3.4.

It is noted that if ρIFR(X) for any non-empty set X, then ρ is an intuitionistic fuzzy symmetric and ∆ρ is an intuitionistic fuzzy irreflexive. Moreover, ρ = ∆ρ ρ (see [4])

Example 3.2. Let n = 3 and ρ = 〈R, Rd〉, σ = 〈S, Sd〉 IFR(I3) be presented by matrices:

  R=\( \left[\begin{array}{ccc}0.5& 0.6& 0.7\\ {}0.7& 0.5& 0.6\\ {}0.6& 0.7& 0.5\end{array}\right],\kern0.5em {R}^d=\left[\begin{array}{ccc}0.4& 0.2& 0.3\\ {}0.3& 0.4& 0.2\\ {}0.2& 0.3& 0.4\end{array}\right] \)

and

$$ S=\left[\begin{array}{ccc}0.6& 0.7& 0.5\\ {}0.5& 0.6& 0.7\\ {}0.7& 0.5& 0.6\end{array}\right],{S}^d=\left[\begin{array}{ccc}0.3& 0.3& 0.4\\ {}0.4& 0.3& 0.3\\ {}0.3& 0.4& 0.3\end{array}\right]. $$

Then, it is easy to see that both ρ and σ are Z3-compatible and

$$ \rho \ominus \sigma =\left[\begin{array}{ccc}\left\langle 0,1\right\rangle & \left\langle \mathrm{0,0.2}\right\rangle & \left\langle \mathrm{0.7,0.3}\right\rangle \\ {}\left\langle \mathrm{0.7,0.3}\right\rangle & \left\langle 0,1\right\rangle & \left\langle \mathrm{0,0.2}\right\rangle \\ {}\left\langle \mathrm{0,0.2}\right\rangle & \left\langle \mathrm{0.7,0.3}\right\rangle & \left\langle 0,1\right\rangle \end{array}\right], $$
$$ \rho \vartriangleright \sigma =\left[\begin{array}{ccc}\left\langle \mathrm{0.5,0.4}\right\rangle & \left\langle \mathrm{0.5,0.4}\right\rangle & \left\langle \mathrm{0.6,0}\right\rangle \\ {}\left\langle \mathrm{0.6,0}\right\rangle & \left\langle \mathrm{0.5,0.4}\right\rangle & \left\langle \mathrm{0.5,0.4}\right\rangle \\ {}\left\langle \mathrm{0.5,0.4}\right\rangle & \left\langle \mathrm{0.6,0}\right\rangle & \left\langle \mathrm{0.5,0.4}\right\rangle \end{array}\right] $$

which they are also Z3-compatible intuitionistic fuzzy relations.

Let ε = 〈E, Ed〉 IFR(In, ) be presented by matrices:

$$ \kern7.5em E=\left[\begin{array}{ccc}0& \cdots & 1\\ {}\vdots & \ddots & \vdots \\ {}1& \cdots & 0\end{array}\right]\ \mathrm{and}\ {E}^d=\left[\begin{array}{ccc}1& \cdots & 0\\ {}\vdots & \ddots & \vdots \\ {}0& \cdots & 1\end{array}\right] $$

Thus,

$$ \varepsilon =\left[\begin{array}{ccc}\left\langle 0,1\right\rangle & \cdots & \left\langle 1,0\right\rangle \\ {}\vdots & \ddots & \vdots \\ {}\left\langle 1,0\right\rangle & \cdots & \left\langle 0,1\right\rangle \end{array}\right] $$

i.e.,

\( \kern7.5em \varepsilon \left(i,j\right)=\left\{\begin{array}{c}\left\langle 1,0\right\rangle \kern0.5em if\ i+j=n+1,\\ {}\left\langle 0,1\right\rangle\ \mathrm{otherwise}.\kern2.75em \end{array}\right. \).

Then, we have the following proposition.

Proposition 3.5. Let ρIFR(In, ) be a Zn-compatible. Then, ερ is an intuitionistic fuzzy symmetric.

Proof. Let τ = ερ. Then, from the definition of the intuitionistic fuzzy relation ε, we can see that τ(i, j) = ρ(n − i + 1, j) for every i, jIn. Since we have that ρ is a Zn-compatible, we get for every i, jIn

τ(i, j) = ρ(n − i + 1 k, jk) (by Proposition 3.2)

and so

$$ \tau \left(i,j\right)=\rho \left(n-i+1\bigoplus k,j\bigoplus k\right)=\rho \left(1,j\bigoplus i\right)={b}_{j\bigoplus i}. $$

Similarly, we can prove that τ(i, j) = bji. Hence, τ is intuitionistic fuzzy symmetric. ☐

Proposition 3.6. Let ρIFR(In, ) be a Zn-compatible. Then, ρ is an intuitionistic fuzzy symmetric if and only if ρ(1, i ) = ρ(1, n + 2 − i ) for every iIn

Proof. Let ρ be intuitionistic fuzzy symmetric. Then,

ρ(1, i )= ρ(k, ik ) = ρ(ik, k ) for every kIn.

Taking k = n − i, we get

$$ \kern0.5em \rho \left(1,i\ \right)=\rho \left(n-1,n-i\ \right)=\rho \left(n,n-i+1\right)=\rho \left(1,n-i+2\ \right). $$

Conversely, suppose ρ(1, i ) = ρ(1, n − i + 2 ) for every iIn. Then, ρ(i, 1) = ρ(ik, k) for every kIn.Taking also k = n − i , we get ρ(i, 1 ) = ρ(1, n − i + 2 ).   Thus, ρ(1, i ) = ρ(i, 1).

Since we have that ρ is a Zn-compatible, we conclude

$$ \rho \left(1,i\ \right)=\rho \left(2,i+1\right)=\rho \left(3,i+2\ \right)=\dots =\rho \left(n,i+n-1\ \right). $$

and

$$ \rho \left(i,1\ \right)=\rho \left(i+1,2\right)=\rho \left(i+2,3\ \right)=\dots =\rho \left(i+n-1,n\ \right). $$

But ρ(1, i ) = ρ(i, 1 ), so that ρ(u, v ) = ρ(v, u ) for every u, vIn. Therefore, ρ is intuitionistic fuzzy symmetric. This completes the proof. ☐

Corollary 3.2.

Let ρIFR(In, ) be a Zn-compatible. Then ρ is an intuitionistic fuzzy symmetric. If and only if ρ(1, i ) = ρ(1, i−1) for every iIn.

Proof. Since i−1 = n − i + 2 (for i ≠ 1 ) with respect to the operation defined on the group (In, ). ☐

Note:

Using Proposition 3.6, the intuitionistic fuzzy relation in Example 3.1 is intuitionistic fuzzy symmetric if and only if b2 = b4 since 2−1 = 4 and 3−1 = 3.

Theorem 3.1.

Let ρ = 〈R, Rd〉, σ = 〈S, Sd〉 IFR(In) be two Zn-compatible. Then, ρσ = σρ.

Proof. Let τ = 〈T, Td〉 = ρσ and η = 〈Q, Qd〉 = σρ. Then, τ and η are Zn-compatible by Proposition 3.3. Now,  τ(1, k) = 〈T(1, k), Td(1, k)〉

$$ =\left\langle \underset{l=1}{\overset{n}{\bigvee \limits }}\left(R\left(1,l\right)\wedge S\left(l,k\right)\right),\underset{l=1}{\overset{n}{\bigwedge \limits }}\left({R}^d\left(1,l\right)\vee {S}^d\right(l,k\Big)\right\rangle $$

and

$$ \eta \left(1,k\right)=\left\langle Q\left(1,k\right),{Q}^d\left(1,k\right)\right\rangle $$
$$ =\left\langle \underset{l=1}{\overset{n}{\bigvee \limits }}\left(S\left(1,l\right)\wedge R\left(l,k\right)\right),\underset{l=1}{\overset{n}{\bigwedge \limits }}\left({S}^d\left(1,l\right)\vee {R}^d\right(l,k\Big)\right\rangle $$

In the following, we show that T(1, k) = Q(1, k) and similarly Td(1, k) = Qd(1, k) ). Since τ and  σ are Zn-compatible intuitionistic fuzzy relations, the fuzzy relations T and Q are also Zn-compatible. We rewrite T(1, k) as follows:

$$ {\displaystyle \begin{array}{l}T\left(1,k\right)={\bigvee}_{l=1}^n\Big(R\left(1,l\right)\wedge S\left(l,k\right)=\\ {}={\bigvee}_{l=1}^k\left(R\left(1,l\right)\wedge S\left(1,k+1-l\right)\right){\bigvee}_{l=k+1}^n\left(R\left(1,l\right)\wedge S\left(l,n+k+1-l\right)\right)\\ {}=\left(R\left(1,1\right)\wedge S\left(1,k\right)\right)\vee \Big(R\left(1,2\right)\wedge S\left(1,k-1\right)\\ {}\vee \left(R\left(1,3\right)\wedge S\left(1,k-2\right)\right)\vee \dots \vee \left(R\left(1,k\right)\wedge S\left(1,1\right)\right)\\ {}\vee \Big(R\left(1,k\right)\wedge S\left(1,1\right)\vee \left(R\left(1,k-1\right)\wedge S\left(1,n\right)\right)\\ {}\vee \Big(R\left(1,k+2\right)\wedge S\left(1,n-2\right)\\ {}\vee \\ {}\vee \left(R\left(1,n-1\right)\wedge S\left(1,k+2\right)\right)\kern0.50em \\ {}\vee \Big(R\left(1,n\right)\wedge S\left(1,k+1\right)\end{array}} $$

Similarly, we compute Q(1, k) as follows:

$$ {\displaystyle \begin{array}{l}Q\left(1,k\right)={\bigvee}_{l=1}^n\Big(S\left(1,l\right)\wedge R\left(l,k\right)\\ {}={\bigvee}_{l=1}^k\left(S\left(1,l\right)\wedge R\left(1,k+1-l\right)\right){\bigvee}_{l=k+1}^n\left(S\left(1,l\right)\wedge R\left(l,n+k+1-l\right)\right)\\ {}=\left(S\left(1,1\right)\wedge R\left(1,k\right)\right)\vee \Big(S\left(1,2\right)\wedge R\left(1,k-1\right)\\ {}\vee \left(S\left(1,3\right)\wedge R\left(1,k-2\right)\right)\vee \dots \vee \left(S\left(1,k\right)\wedge R\left(1,1\right)\right)\\ {}\vee \Big(S\left(1,k\right)\wedge R\left(1,1\right)\vee \left(S\left(1,k-1\right)\wedge R\left(1,n\right)\right)\\ {}\vee \left(S\left(1,k+2\right)\wedge R\left(1,n-2\right)\right)\\ {}\vee \\ {}\vee \left(S\left(1,n-1\right)\wedge R\left(1,k+2\right)\right)\\ {}\vee \Big(S\left(1,n\right)\wedge R\left(1,k+1\right)\end{array}} $$

From these computations of T(1, k) and Q(1, k), it is clear that =Q(1, k). In a similar way, we can show that Td(1, k) = Qd(1, k). Thus, τ(1, k) = η(1, k). Again, since τ and η are Zn-compatible intuitionistic fuzzy relations and τ(1, k) = η(1, k), we get τ =η, i.e., ρσ = σρ.

Definition 3.2. For ρ = 〈R, Rd〉 IFR(In), we define the two intuitionistic fuzzy relations ρL = 〈RL, (Rd)Land ρU = 〈RU, (Rd)Uas follows:

$$ {R}_L\left(i,j\right)=\min \left\{R\left(u\ast i\ast k\ast v,u\ast j\ast k\ast v\right),\kern1.25em i,j,k,u,v\in {I}_n\right\}, $$
$$ {\left({R}^d\right)}_L\left(i,j\right)=\max \left\{{R}^d\left(u\ast i\ast k\ast v,u\ast j\ast k\ast v\right),\kern1.25em i,j,k,u,v\in I\_n\ \right\} $$

and

$$ {R}_U\left(i,j\right)=\max \left\{R\left(u\ast i\ast k\ast v,u\ast j\ast k\ast v\right),\kern1.25em i,j,k,u,v\in {I}_n\right\}, $$
$$ {\left({R}^d\right)}_U\left(i,j\right)=\min \left\{{R}^d\left(u\ast i\ast k\ast v,u\ast j\ast k\ast v\right),\kern1.25em i,j,k,u,v\in I\_n\ \right\} $$

The following two propositions have interesting properties. They enable us to construct two Zn-compatible intuitionistic fuzzy relations ρL and ρU on (In, ) from any given intuitionistic fuzzy relation ρ on (In, ). Moreover, ρL ≤ ρ ≤ ρU.

Proposition 3.7. Let ρ = 〈R, Rd〉 IFR(In). Then, ρLIFR(In, ) is a Zn-compatible intuitionistic fuzzy relation. Moreover, it is the largest one contained in ρ.

Proof. First, we notice that the number ukv must be one of the different elements of the group (In, ) and so also uv. Now, we show that the intuitionistic fuzzy relation ρL is a Zn-compatible. We have:

$$ {R}_L\left(i,j\right)=\min \left\{R\left(u\ast i\ast v,u\ast j\ast v\right),\kern1.25em i,j,u,v\in {I}_n\right\} $$
$$ \kern0.75em =\min \left\{R\left(i\ast 1,j\ast 1\right),\Big(R\left(i\ast 2,j\ast 2\right),\dots, \Big(R\left(i\ast n,j\ast n\right)\right\}. $$

Also,

$$ {R}_L\left(i\ast k,j\ast k\right)=\min \left\{R\left(u\ast i\ast k\ast v,u\ast j\ast k\ast v\right),\kern0.5em i,j,k,u,v\in {I}_n\right\} $$
$$ \kern0.5em =\min \left\{R\left(i\ast 1,j\ast 1\right),R\left(i\ast 2,j\ast 2\right),\dots, R\left(i\ast n,j\ast n\right)\right\}. $$

Therefore, RL(i, j) = RL(ik, jk). Hence, the fuzzy relation RL is a Zn-compatible fuzzy relation. Similarly, we can prove that the fuzzy relation (Rd)L is a Zn-compatible fuzzy relation. Therefore, ρL is a Zn-compatible intuitionistic fuzzy relation.

By the definition of ρL it is clear that RL(i, j) ≤ R(i, j) and also, (Rd)L(i, j) ≥ Rd(i, j) Thus, ρL ≤ ρ.

Now, let ξ = 〈L, Ld〉 IFR(In, ) be a Zn-compatible such that ξ ≤ ρ. We will show that ξ ≤ ρL. Since we have that

L(i, j) = L(ui, uj) = L(uiv, ujv) ≤ R(uiv, ujv) for every i, j, u, vIn, we obtain

$$ L\left(i,j\right)\le \mathit{\min}\left\{R\left(u\ast i\ast v,u\ast j\ast v\right):\kern0.5em i,j,u,v\in {I}_n\ \right\}={R}_L\left(i,j\right). $$

Similarly, we can show that

$$ {L}^d\left(i,j\right)\ge \max \left\{{R}^d\left(u\ast i\ast v,u\ast j\ast v\right):\kern0.5em i,j,u,v\in {I}_n\ \right\}={\left({R}^d\right)}_L\left(i,j\right). $$

Thus, ξ ≤ ρL. ☐

Proposition 3.8: Let \( \rho =\left\langle R,{R}^d\right\rangle, \kern0.5em {\rho}_1=\left\langle {R}_1,{R}_1^d\right\rangle, \kern0.5em {\rho}_2=\left\langle {R}_{2,}{R}_2^d\right\rangle \in IFR\left({I}_n,\ast \right). \) Then, we have:

  1. (1)
    $$ {\rho}_1\le {\rho}_2\ \mathrm{implies}\ {\left({\rho}_1\right)}_U\le {\left({\rho}_2\right)}_U $$
  2. (2)
    $$ {\rho}_U\ \mathrm{is}\ \mathrm{a}\ {Z}_n-\mathrm{compatible} $$
  3. (3)
    $$ {\left({\rho}_U\right)}^{-1}={\left({\rho}^{-1}\right)}_U $$
  4. (4)
    $$ {\left({\rho}_1\bigvee {\rho}_2\right)}_{\Big)U}={\left({\rho}_1\right)}_U\bigvee {\left({\rho}_2\right)}_U $$

Proof. The proof of (1) is clear and the proof of (2) is similar to the proof of ρL in Proposition 3.7. However, we only prove (3) and (4).

(3) Let ρU = 〈RU, (Rd)U〉 IFR(In, ). Then,

$$ {\left({R}_U\right)}^{-1}\left(i,j\right)={R}_U\left(j,i\right)=\max \left\{R\left(u\ast j\ast v,u\ast i\ast v\right):\kern0.5em i,j,u,v\in {I}_n\right\} $$
$$ =\max \left\{{R}^{-1}\left(u\ast j\ast v,u\ast i\ast v\right):\kern0.5em i,j,u,v\in {I}_n\right\} $$
$$ ={\left({R}^{-1}\right)}_U $$

Thus,

$$ {\left({R}_U\right)}^{-1}={\left({R}^{-1}\right)}_U $$

Similarly, we can show that ((Rd)U)−1 = ((Rd)−1)U. This completes the proof and hence (ρU)−1 = (ρ−1)U

(4) \( {\left({\rho}_1\right)}_U\vee {\left({\rho}_2\right)}_U=\left\langle \left({\left({R}_1\right)}_U\vee {\left({R}_2\right)}_U\right),\Big({\left({R}_1^d\right)}_U\wedge {\left({R}_2^d\right)}_U\right\rangle \), and \( {\left({\rho}_1\vee {\rho}_2\right)}_U=\left\langle {\left({R}_1\vee {R}_2\right)}_U,{\left({R}_1^d\wedge {R}_2^d\right)}_U\right\rangle . \)

Now, we show that ((R1)U (R2)U) = (R1R2)U. For every i, j, u, vIn, we have:

$$ \left({\left({R}_1\right)}_U\vee {\left({R}_2\right)}_U\right)\left(i,j\right)=\max \left({\left({R}_1\right)}_U\left(i,j\right),{\left({R}_2\right)}_U\left(i,j\right)\right) $$
$$ =\max \left(\max \left\{{R}_1\left(u\ast i\ast v,u\ast j\ast v\right)\right\},\max \left\{{R}_2\left(u\ast i\ast v,u\ast j\ast v\right)\right\}\right) $$
$$ =\max \left(\left\{\max \Big({R}_1\left(u\ast i\ast v,u\ast j\ast v\right)\right\},\left\{{R}_2\left(u\ast i\ast v,u\ast j\ast v\right)\right\}\right) $$
$$ =\mathit{\max}\left\{\left({R}_1\vee {R}_2\right)\Big(u\ast i\ast v,u\ast j\ast v\right\}={\left({R}_1\vee {R}_2\right)}_U\left(i,j\right). $$

Also, for every i, j, u, vIn we have:

(\( {\left({R}_1^d\right)}_U\wedge {\left({R}_2^d\right)}_U\left)\left(i,j\right)=\min \right({\left({R}_1^d\right)}_U\left(i,j\right),{\left({R}_2^d\right)}_U\left(i,j\right) \)

$$ =\min \left(\min \left\{\left({R}_1^d\right(u\ast i\ast k\ast v,u\ast j\ast v\right\},\min \left\{\left({R}_2^d\right(u\ast i\ast k\ast v,u\ast j\ast k\ast v\right\}\right) $$
$$ =\min \left(\left\{\min \left({R}_1^d\right(u\ast i\ast v,u\ast j\ast v\right\},\left\{\left({R}_2^d\right(u\ast i\ast v,u\ast j\ast v\right\}\right) $$
$$ =\min \left\{\left({R}_1^d\wedge {R}_2^d\right)\Big(u\ast i\ast v,u\ast j\ast v\Big)\right\}={\left({R}_1^d\wedge {R}_2^d\right)}_U\left(i,j\right). $$

Therefore, (ρ1ρ2)U = (ρ1)U (ρ2)U. ☐

Analogously, we can prove the following proposition.

Proposition 3.9. Let \( \rho =\left\langle R,{R}^d\right\rangle, \kern0.5em {\rho}_1=\left\langle {R}_1,{R}_1^d\right\rangle, \kern0.5em {\rho}_2=\left\langle {R}_{2,}{R}_2^d\right\rangle \in IFR\left({I}_n,\ast \right). \) Then, we have:

  1. (1)

    ρ1 ≤ ρ2 implies (ρ1)L ≤ (ρ2)L

  2. (2)

    (ρL)−1 = (ρ−1)L

  3. (3)

    (ρ1ρ2)L = (ρ1)L (ρ2)L

Corollary 3.3. Let ρ be intuitionistic reflexive (irreflexive). Then, ρU (ρL) is intuitionistic fuzzy reflexive (irreflexive).

Proposition 3.10. Let ρ = 〈R, Rd〉 IFR(In, ). If ρ is a Zn-compatible, then ρL = ρU = ρ.

Proof Let ρU = 〈RU, (Rd)U〉 be a Zn-compatible intuitionistic fuzzy relation. Then, both RU and (Rd)U are also Zn-compatible fuzzy relations and so,

$$ {R}_U\left(i,j\right)=\max \left\{\Big(R\left(u\ast i\ast v,u\ast j\ast v\right),\kern1.25em i,j,u,v\in {I}_n\right\}=R\left(i,j\right) $$

and

$$ {\left({R}^d\right)}_U\left(i,j\right)=\min \Big\{\left({R}^d\left(u\ast i\ast v,u\ast j\ast v\right),\kern1.25em i,j,u,v\in {I}_n\right\}=R\left(i,j\right) $$

Therefore, ρU = ρ.

Similarly, it can be shown that ρL = ρ. ☐

Example 3.3: For n = 4, let ρ = 〈R, Rd〉 IFR(I4) be presented by matrices:

$$ R=\left[\begin{array}{cccc}0.5& 0.7& 0.8& 0.2\\ {}0.8& 0.3& 0.4& 0.9\\ {}0.7& 0.5& 0.3& 0.6\\ {}0.9& 1.0& 0& 0.4\end{array}\right]\ \mathrm{and}\ {R}^d=\left[\begin{array}{cccc}0.5& 0.3& 0.2& 0.6\\ {}0.1& 0.6& 0.6& 0\\ {}0.3& 0.4& 0.5& 0.4\\ {}0.1& 0& 1.0& 0.3\end{array}\right] $$

Thus,

$$ \rho =\left[\begin{array}{cccc}\left\langle 0.5,0.5\right\rangle & \left\langle 0.7,0.3\right\rangle & \left\langle 0.8,0.2\right\rangle & \left\langle 0.2,0.6\right\rangle \\ {}\left\langle 0.8,0.1\right\rangle & \left\langle 0.3,0.6\right\rangle & \left\langle 0.4,0.6\right\rangle & \left\langle 0.9,0\right\rangle \\ {}\left\langle 0.7,0.3\right\rangle & \left\langle 0.5,0.4\right\rangle & \left\langle 0.3,0.5\right\rangle & \left\langle 0.6,0.4\right\rangle \\ {}\left\langle 0.9,0.1\right\rangle & \left\langle 1,0\right\rangle & \left\langle 0,1\right\rangle & \left\langle 0.4,0.3\right\rangle \end{array}\right] $$

Since

$$ {R}_U=\left[\begin{array}{cccc}0.5& 0.9& 1& 0.8\\ {}0.8& 0.5& 0.9& 1\\ {}1& 0.8& 0.5& 0.9\\ {}0.9& 1& 0.8& 0.5\end{array}\right]\ \mathrm{and}\ {\left({R}^d\right)}_U=\left[\begin{array}{cccc}0.3& 0.1& 0.0& 0.1\\ {}0.1& 0.3& 0.1& 0.0\\ {}0.0& 0.1& 0.3& 0.1\\ {}0.1& 0.0& 0.1& 0.3\end{array}\right] $$
$$ {R}_L=\left[\begin{array}{cccc}0.3& 0.4& 0.7& 0\\ {}0& 0.3& 0.4& 0.7\\ {}0.7& 0& 0.3& 0.4\\ {}0.4& 0.7& 0& 0.3\end{array}\right]\ \mathrm{and}\ {\left({R}^d\right)}_L=\left[\begin{array}{cccc}0.6& 0.6& 0.3& 1\\ {}1& 0.6& 0.6& 0.3\\ {}0.3& 1& 0.6& 0.6\\ {}0.6& 0.3& 1& 0.6\end{array}\right], $$

we have:

$$ {\rho}_U=\left[\begin{array}{cccc}\left\langle 0.5,0.3\right\rangle & \left\langle 0.9,0.1\right\rangle & \left\langle 1,0\right\rangle & \left\langle 0.8,0,1\right\rangle \\ {}\left\langle 0.8,0,1\right\rangle & \left\langle 0.5,0.3\right\rangle & \left\langle 0.9,0.1\right\rangle & \left\langle 1,0\right\rangle \\ {}\left\langle 1,0\right\rangle & \left\langle 0.8,0,1\right\rangle & \left\langle 0.5,0.3\right\rangle & \left\langle 0.9,0.1\right\rangle \\ {}\left\langle 0.9,0.1\right\rangle & \left\langle 1,0\right\rangle & \left\langle 0.8,0,1\right\rangle & \left\langle 0.5,0.3\right\rangle \end{array}\right] $$

and

$$ {\rho}_L=\left[\begin{array}{cccc}\left\langle 0.3,0.6\right\rangle & \left\langle 0.4,0.6\right\rangle & \left\langle 0.7,0.3\right\rangle & \left\langle 0,1\right\rangle \\ {}\left\langle 0,1\right\rangle & \left\langle 0.3,0.6\right\rangle & \left\langle 0.4,0.6\right\rangle & \left\langle 0.7,0.3\right\rangle \\ {}\left\langle 0.7,0.3\right\rangle & \left\langle 0,1\right\rangle & \left\langle 0.3,0.6\right\rangle & \left\langle 0.4,0.6\right\rangle \\ {}\left\langle 0.4,0.6\right\rangle & \left\langle 0.7,0.3\right\rangle & \left\langle 0,1\right\rangle & \left\langle 0.3,0.6\right\rangle \end{array}\right]. $$

It is clear that the fuzzy relations RU, (Rd)U, RL, and (Rd)L are all Z4-compatible and so ρU and ρL are Z4-compatible intuitionistic fuzzy relations. Also, we see that RL ≤ R ≤ RU and (Rd)L ≥ R ≥ (Rd)U. So that ρL ≤ ρ ≤ ρU .

As an instance, we compute RU(2, 3) and (Rd)U(2, 3) as follows:

RU(2, 3) = max {R(u 2 v, u 3 v) :   u, vI4}. Since we have that (I4, ) is an abelian group, we conclude uv must be one of the numbers 1, 2, 3 and 4. So that

$$ {R}_U\left(2,3\right)=\max \left\{R\left(2\ast 1,3\ast 1\right),R\left(2\ast 2,3\ast 2\right),R\left(2\ast 3,3\ast 3\right),R\left(2\ast 4,3\ast 4\right)\right\} $$
$$ =\max \left\{R\left(2,3\right),R\left(3,4\right),R\left(4,1\right),R\left(1,2\right)\right\} $$
$$ =\max \left\{0.4,0.6,0.9,0.7\right\}=0.9 $$

Also

$$ {\left({R}^d\right)}_U\left(2,3\right)=\min \left\{{R}^d\left(2,3\right),{R}^d\left(3,4\right),{R}^d\left(4,1\right),{R}^d\left(1,2\right)\right\}=0 $$

Then

$$ {\rho}_U\left(2,3\right)=\left\langle 0.9,0.1\right\rangle $$
$$ \mathrm{Similarly},\mathrm{we}\ \mathrm{have}\ {\rho}_L\left(2,3\right)=\left\langle 0.4,0.6\right\rangle $$

Conclusion

The operations ,   ,  and play an important role in the whole of the paper. We explored the interesting properties of intuitionistic fuzzy relations which are compatible with the group (Zn, ) which isomorphic to the group (In, ). These relations are, of course, finite and they are circulants. Given any finite intuitionistic fuzzy relation ρ, we define in Section 3 two finite intuitionistic fuzzy relations ρL and ρU. We showed that these relations are compatible with the group (Zn, ), and hence they are circulants. Thus, this result enables us to construct two circulant intuitionistic fuzzy relations from any given one.

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Abbreviations

IFR(X):

The family of intuitionistic fuzzy relations on a nonempty set X

IFR(In, ):

The family of intuitionistic fuzzy relations on the group (In, ), where In = {1, 2, …, n}

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Emam, E.G. Intuitionistic fuzzy relations compatible with the group Zn. J Egypt Math Soc 27, 54 (2019). https://doi.org/10.1186/s42787-019-0053-6

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Keywords

  • Intuitionistic fuzzy relations
  • Fuzzy relations
  • Compatible fuzzy relations

2010 Mathematics Subject Classification

  • 15B15
  • 15B33