Intuitionistic Fuzzy Ideals of Ternary Near-Rings

Warud Nakkhasen

doi:10.5391/IJFIS.2020.20.4.290

International Journal of Fuzzy Logic and Intelligent Systems 2020;20(4):290-297

Published online December 25, 2020

Warud Nakkhasen

Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, Thailand

Correspondence to: Warud Nakkhasen (warud.n@msu.ac.th)

Received August 7, 2018; Revised November 4, 2020; Accepted November 23, 2020.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract: We define the concept of intuitionistic fuzzy ideals of ternary near-rings as a generalization of fuzzy ideals, and we investigate some of their properties. Moreover, we characterize the notions of Noetherian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.; Keywords : Ternary near-ring, Fuzzy ideal, Intuitionistic fuzzy ideal, Intuitionistic fuzzy set

1. Introduction

The concept of fuzzy sets was first introduced by Zadeh [1] as a function of a nonempty set X on the unit interval [0, 1]. The first inspired application to many algebraic structures was the concept of fuzzy groups, introduced by Rosenfeld [2]. Lui [3] studied the fuzzy ideals of rings, and many researchers [4–6] have extended these concepts. The notions of fuzzy subnear-rings and fuzzy left (resp. right) ideals in near-rings were introduced by Abou-Zaid in [7]. They have been studied by many authors [8–10].

The concept of intuitionistic fuzzy sets was introduced by Atanassov [11–13] as a generalization of the concept of fuzzy sets. Fuzzy sets give the degree of membership of an element in a given set. Intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership. This theory has been studied by many mathematicians [14–19]. Biswas [20] considered the notion of intuitionistic fuzzy subgroups of groups. In [21], the authors presented the concept of intuitionistic fuzzy ideals of semi-rings. Later, the concept of intuitionistic fuzzy ideals of near-rings was introduced and studied by Zhan and Ma [22].

In 2012, Nakkhasen and Pibaljomme [23] introduced the concept of left ternary near-rings and investigated some properties of L-fuzzy ideals of ternary near-rings, where L is a complete lattice with the greatest element 1 and the least element 0. Later, Uma Maheswari and Meera [24, 25] studied the concepts of fuzzy soft ideals and fuzzy soft prime ideals over right ternary near-rings. In this paper, we introduce the notion of intuitionistic fuzzy ideals of ternary near-rings and investigate some of their properties. Also, we introduce and characterize the notions of Noetherian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.

2. Preliminaries

In this section, we present the basic definitions that are used in the following sections of this paper.

Definition 1 [26]

A ternary semi-group is an algebraic structure (N,+, [ ]) such that N is a nonempty set, and [ ] : N³ → N is a ternary operation satisfying the following associative law: [[abc]de] = [a[bcd]e] = [ab[cde]], for all a, b, c, d, e ∈ N.

Definition 2 [26]

Let A, B, and C be nonempty subsets of a ternary near-ring N. Then, [ABC] = {[abc] ∈ N | a ∈ A, b ∈ B, c ∈ C}.

Definition 3 [23]

Let N be a nonempty set together with a binary operation + and a ternary operation [ ] : N³ → N. Then, (N,+, [ ]) is called a left ternary near-ring if it satisfies the following conditions:

(i) (N, +) is a group (not necessarily abelian);
(ii) (N, [ ]) is a ternary semi-group;
(iii) [ab(c + d)] = [abc] + [abd], for every a, b, c, d ∈ N.

Right ternary near-rings and lateral ternary near-rings are defined in a similar manner. In this paper, we focus on left ternary near-rings, and we will use the word “ternary near-rings” to mean “left ternary near-rings.”

Definition 4 [23]

A nonempty subset T of a ternary near-ring N is said to be a ternary subnear-ring of N if a − b ∈ T and [abc] ∈ T, for all a, b, c ∈ T.

Definition 5 [24]

Let N be a nonempty near-ring. Let I be a normal subgroup of (N, +). Then, for every a, b, c ∈ N and i ∈ I,

(i) I is called a left ideal of N if [NNI] ⊆ I;
(ii) I is called a right ideal of N if [(a + i)bc] − [abc] ∈ I;
(iii) I is called a lateral ideal of N if [a(b + i)c] − [abc] ∈ I.

We call I an ideal of N if it is a left ideal, a right ideal, and a lateral ideal of N.

Example 1 [23]

Let N = {a, b, c, d} be a set with a binary operation + on N as follows:

+	a	b	c	d
a	a	b	c	d
b	b	a	d	c
c	c	d	b	a
d	d	c	a	b

The ternary operation [ ] on N is defined by [xyz] = z for all x, y, z ∈ N. Then, we have that (N,+, [ ]) is a ternary near-ring. Let I = {a, b}. It follows that I is a ternary subnear-ring of N. Next, we show that I is a left ideal, a right ideal, and a lateral ideal of N, that is, I is an ideal of N.

Definition 6 [23]

Let N and R be ternary near-rings. A mapping ϕ : N → R is called a homomorphism if ϕ(a + b) = ϕ(a) + ϕ(b) and ϕ([abc]) = [ϕ(a)ϕ(b)ϕ(c)], for all a, b, c ∈ N.

Let X be a nonempty set. A fuzzy set [1] of X is a mapping μ : X → [0, 1]. Let μ be a fuzzy set of X. The set U(μ; t) = {x ∈ X | μ(x) ≥ t} is called an upper-level set of μ, and the set L(μ; t) = {x ∈ X | μ(x) ≤ t} is called a lower-level set of μ, where t ∈ [0, 1]. The complement of μ denoted by μ^c is the fuzzy set of X defined by μ^c(x) = 1 − μ(x), for all x ∈ X. The intersection and union of two fuzzy sets μ and λ of X, denoted by μ ∩ λ and μ ∪ λ, respectively, are defined by letting x ∈ X, (μ ∩ λ)(x) = min{μ(x), λ(x)} and (μ ∪ λ)(x) = max{μ(x), λ(x)}, respectively.

The concept of intuitionistic fuzzy sets was introduced by Atanassov [11–13] as an important generalization of the concept of fuzzy sets. An intuitionistic fuzzy set A in a nonempty set X is defined by the form

A = {(x, μ_{A} (x), λ_{A} (x)) | x \in X},

where μ_A : X → [0, 1] and λ_A : X → [0, 1] denote the degree of membership and the degree of non-membership of each x ∈ X in the set A, and also 0 ≤ μ_A(x) + λ_A(x) ≤ 1, for all x ∈ X. For the sake of convenience, we will use the symbol A = (μ_A, λ_A) instead of the intuitionistic fuzzy set A = {(x, μ_A(x), λ_A(x)) | x ∈ X}.

3. Intuitionistic Fuzzy Ideals

In this section, we introduce the concept of intuitionistic fuzzy ideals of ternary near-rings and investigate some of their properties.

Definition 7

An intuitionistic fuzzy set A = (μ_A, λ_A) of a ternary near-ring (N,+, [ ]) is called an intuitionistic fuzzy ideal of N if it satisfies for every i, x, y, z ∈ N,

(IF1) μ_A(x − y) ≥ min{μ_A(x), μ_A(y)};
(IF2) μ_A([xyz]) ≥ min{μ_A(x), μ_A(y), μ_A(z)};
(IF3) μ_A(y + x − y) ≥ μ_A(x);
(IF4) μ_A([xyz]) ≥ μ_A(z);
(IF5) μ_A([(x + i)yz] − [xyz]) ≥ μ_A(i);
(IF6) μ_A([x(y + i)z] − [xyz]) ≥ μ_A(i);
(AF1) λ_A(x − y) ≤ max{λ_A(x), λ_A(y)};
(AF2) λ_A([xyz]) ≤ max{λ_A(x), λ_A(y), λ_A(z)};
(AF3) λ_A(y + x − y) ≤ λ_A(x);
(AF4) λ_A([xyz]) ≤ λ_A(z);
(AF5) λ_A([(x + i)yz] − [xyz]) ≤ λ_A(i);
(AF6) λ_A([x(y + i)z] − [xyz]) ≤ λ_A(i).

Example 2

In Example 1, we define an intuitionistic fuzzy set A = (μ_A, λ_A) of a ternary near-ring N by μ_A(c) = μ_A(d) < μ_A(b) < μ_A(a) and λ_A(a) < λ_A(b) < λ_A(c) = λ_A(d). By routine calculations, it is clear that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N.

Proposition 1

If A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of a ternary near-ring N, then μ_A(0) ≥ μ_A(x) and λ_A(0) ≤ λ_A(x), for all x ∈ N.

Proof

Assume that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of a ternary near-ring N. Let x ∈ N. Then, μ_A(0) = μ_A(x − x) ≥ min{μ_A(x), μ_A(x)} = μ_A(x) and λ_A(0) = λ_A(x − x) ≤ max{λ_A(x), λ_A(x)} = λ_A(x).

Theorem 1

Let N be a ternary near-ring. Then A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N if and only if for any t, s ∈ [0, 1], the nonempty sets U(μ_A; t) and L(λ_A; s) are ideals of N.

Proof

Assume that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. Let s, t ∈ [0, 1]. First, let x, y ∈ U(μ_A; t). Then, μ_A(x) ≥ t and μ_A(y) ≥ t. Thus, μ_A(x − y) ≥ min{μ_A(x), μ_A(y)} ≥ t, and so x − y ∈ U(μ_A; t). Second, for any x ∈ U(μ_A; t) and n ∈ N, we have μ_A(n + x − n) ≥ μ_A(x) ≥ t, and then n + x − n ∈ U(μ_A; t). Third, for any n, m ∈ N and x ∈ U(μ_A; t), we have μ_A([nmx]) ≥ μ_A(x) ≥ t, that is, [nmx] ∈ U(μ_A; t). Finally, for any i ∈ U(μ_A; t) and x, y, z ∈ N. Then, μ_A([(x + i)yz] − [xyz]) ≥ μ_A(i) ≥ t and μ_A([x(y + i)z] − [xyz]) ≥ μ_A(i) ≥ t. It follows that [(x + i)yz] − [xyz] ∈ U(μ_A; t) and [x(y + i)z] − [xyz] ∈ U(μ_A; t). Hence, U(μ_A; t) is an ideal of N. Similarly, we can show that L(λ_A; s) is also an ideal of N.

Conversely, assume that for any for any s, t ∈ [0, 1], the nonempty sets U(μ_A; t) and L(λ; s) are ideals of N. We want to show that μ_A satisfies (IF1)–(IF6) and λ_A satisfies (AF1)–(AF6). Let i, x, y, z ∈ N. (IF1) Let t₁ ∈ [0, 1] such that t₁ = min{μ_A(x), μ_A(y)}. Then, μ_A(x) ≥ t₁ and μ_A(y) ≥ t₁, so x, y ∈ U(μ_A; t₁). By assumption, x−y ∈ U(μ_A; t₁). Thus, μ_A(x − y) ≥ t₁ = min{μ_A(x), μ_A(y)}. (IF2) Let t₂ ∈ [0, 1] such that t₂ = min{μ_A(x), μ_A(y), μ_A(z)}. Then, μ_A(x) ≥ t₂, μ_A(y) ≥ t₂, μ_A(z) ≥ t₂, that is, x, y, z ∈ U(μ_A; t₂). By assumption, [xyz] ∈ U(μ_A; t₂). Thus, μ_A([xyz]) ≥ t₂ = min{μ_A(x), μ_A(y), μ_A(z)}. (IF3) Let t₃ ∈ [0, 1] such that t₃ = μ_A(x). Then x ∈ U(μ_A; t₃). So, y + x − y ∈ U(μ_A; t₃). Hence, μ_A(y + x − y) ≥ t₃ = μ_A(x). (IF4) Let t₄ ∈ [0, 1] such that t₄ = μ_A(z). Then z ∈ U(μ_A; t₄). Thus, [xyz] ∈ U(μ_A; t₄), and then μ_A([xyz]) ≥ t₄ = μ_A(z). (IF5) Let t₅ ∈ [0, 1] such that t₅ = μ_A(i). Then i ∈ U(μ_A; t₅). Thus, [(x + i)yz] − [xyz] ∈ U(μ_A; t₅), that is, μ_A([(x + i)yz] − [xyz]) ≥ t₅ = μ_A(i). (IF6) Let t₆ ∈ [0, 1] such that t₆ = μ_A(i). Then, i ∈ U(μ_A; t₆). Thus, [x(y+i)z]− [xyz] ∈ U(μ_A; t₆). It follows that μ_A([x(y + i)z] − [xyz]) ≥ t₆ = μ_A(i). Therefore, μ_A satisfies (IF1)–(IF6). Similarly, we can prove that λ_A satisfies (AF1)–(AF6). This completes the proof.

Example 3

In Example 2, it is clear that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. Then, we can show that the upper-level sets of N are U(μ_A; μ_A(a)) = {a}, U(μ_A; μ_A(b)) = {a, b}, U(μ_A; μ_A(c)) = N, and U(μ_A; μ_A(d)) = N. Also, the lower-level sets of N are L(λ_A; λ_A(a)) = {a}, L(λ_A; λ_A(b)) = {a, b}, L(λ_A; λ_A(c)) = N, and L(λ_A; λ_A(d)) = N. By Theorem 1, it follows that {a}, {a, b}, and N are the ideals of N.

Theorem 2

Let N be a ternary near-ring. Then, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N if and only if $A^{c} = (λ_{A}^{c}, μ_{A}^{c})$ is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. Let i, x, y, z ∈ N. (IF1) $λ_{A}^{c} (x - y) = 1 - λ_{A} (x - y) \geq 1 - max {λ_{A} (x), λ_{A} (y)} = min {1 - λ_{A} (x), 1 - λ_{A} (y)} = min {λ_{A}^{c} (x), λ_{A}^{c} (y)}$ . Similarly, we can show that $λ_{A}^{c}$ satisfies (IF2)–(IF6). Next, we consider (AF1) $μ_{A}^{c} (x - y) = 1 - μ_{A} (x - y) \leq 1 - min {μ_{A} (x), μ_{A} (y)} = max {1 - μ_{A} (x), 1 - μ_{A} (y)} = max {μ_{A}^{c} (x), μ_{A}^{c} (y)}$ . Similarly, we can show that (AF2)–(AF6). Hence, $A^{c} = (λ_{A}^{c}, μ_{A}^{c})$ is an intuitionistic fuzzy ideal of N.

Conversely, assume that $A^{c} = (λ_{A}^{c}, μ_{A}^{c})$ is an intuitionistic fuzzy ideal of N. Let i, x, y, z ∈ N. (IF1) $1 - μ_{A} (x - y) = μ_{A}^{c} (x - y) \leq max {μ_{A}^{c} (x), μ_{A}^{c} (y)} = max {1 - μ_{A} (x), 1 - μ_{A} (y)} = 1 - min {μ_{A} (x), μ_{A} (y)}$ . This implies that μ_A(x − y) ≥ min{μ_A(x), μ_A(y)}. Similarly, we can show that μ_A satisfies (IF2)–(IF6). Next, we consider (AF1) $1 - λ_{A} (x - y) = λ_{A}^{c} (x - y) \geq min {λ_{A}^{c} (x), λ_{A}^{c} (y)} = min {1 - λ_{A} (x), 1 - λ_{A} (y)} = 1 - max {λ_{A} (x), λ_{A} (y)}$ . It follows that λ_A(x − y) ≤ max{λ_A(x), λ_A(y)}. Similarly, we can show that λ_A satisfies (AF2)–(AF6). Therefore, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N.

The following theorem immediately follows from Theorem 2.

Theorem 3

Let A = (μ_A, λ_A) be an intuitionistic fuzzy set of a ternary near-ring N. Then, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N if and only if $A^{μ} = (μ_{A}, μ_{A}^{c})$ and $A^{λ} = (λ_{A}^{c}, λ_{A})$ are intuitionistic fuzzy ideals of N.

Theorem 4

If A = (μ_A, λ_A) and B = (μ_B, λ_B) are intuitionistic fuzzy ideals of a ternary near-ring N, then A ∪ B = (μ_A ∪ μ_B, λ_A ∩ λ_B) is also an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μ_A, λ_A) and B = (μ_B, λ_B) are intuitionistic fuzzy ideals of a ternary near-ring N. Clearly, A ∪ B is an intuitionistic fuzzy set of N. Let i, x, y, z ∈ N. (IF1) Then,

\begin{array}{l} (μ_{A} \cup μ_{B}) (x - y) = max {μ_{A} (x - y), μ_{A} (x - y)} \\ \geq max {min {μ_{A} (x), μ_{A} (y)}, min {μ_{B} (x), μ_{B} (y)}} \\ = min {max {μ_{A} (x), μ_{B} (x)}, max {μ_{A} (y), μ_{B} (y)}} \\ = min {(μ_{A} \cup μ_{B}) (x), (μ_{A} \cup μ_{B}) (y)} . \end{array}

The proofs of (IF2)–(IF6) are similar to that of (IF1). Next, (AF1) we have

\begin{array}{l} (λ_{A} \cap λ_{B}) (x - y) = min {λ_{A} (x - y), λ_{B} (x - y)} \\ \leq min {max {λ_{A} (x), λ_{A} (y)}, max {λ_{B} (x), λ_{B} (y)}} \\ = max {min {λ_{A} (x), λ_{B} (x)}, min {λ_{A} (y), λ_{B} (y)}} \\ = max {(λ_{A} \cap λ_{B}) (x), (λ_{A} \cap λ_{B}) (y)} . \end{array}

The proofs of (AF2)–(AF6) are similar to that of (AF1). Therefore, A ∪ B is an intuitionistic fuzzy ideal of N.

Theorem 5

If A = (μ_A, λ_A) and B = (μ_B, λ_B) are intuitionistic fuzzy ideals of a ternary near-ring N, then A ∩ B = (μ_A ∩ μ_B, λ_A ∪ λ_B) is also an intuitionistic fuzzy ideal of N.

Proof

The proof is similar to Theorem 4.

Let f : N → R be a homomorphism of ternary near-rings. For any A = (μ_A, λ_A) of R, we define a new $A^{f} = (μ_{A}^{f}, λ_{A}^{f})$ of N by $μ_{A}^{f} (x) = μ_{A} (f (x))$ and $λ_{A}^{f} (x) = λ_{A} (f (x))$ , for all x ∈ N.

Theorem 6

Let f : N → R be a homomorphism of ternary near-rings. If A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of R, then $A^{f} = (μ_{A}^{f}, λ_{A}^{f})$ is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of R. Let i, x, y, z ∈ N. Then, (IF1) $μ_{A}^{f} (x - y) = μ_{A} (f (x - y)) = μ_{A} (f (x) - f (y)) \geq min {μ_{A} (f (x)), μ_{A} (f (y))} = min {μ_{A}^{f} (x), μ_{A}^{f} (y)}$ . Similarly, we can prove that $μ_{A}^{f}$ satisfies (IF2)–(IF6). Now, (AF1) $λ_{A}^{f} (x - y) = λ_{A} (f (x - y)) = λ_{A} (f (x) - f (y)) \leq max {λ_{A} (f (x)), λ_{A} (f (y))} = max {λ_{A}^{f} (x), λ_{A}^{f} (y)}$ . Similarly, we can prove that $λ_{A}^{f}$ satisfies (AF2)–(AF6). Therefore, $A^{f} = (μ_{A}^{f}, λ_{A}^{f})$ is an intuitionistic fuzzy ideal of N.

If we strengthen the condition of f, we can construct the converse of Theorem 6 as follows.

Theorem 7

Let f : N → R be an epimorphism of ternary near-rings, and let A = (μ_A, λ_A) be an intuitionistic fuzzy set of R. If $A^{f} = (μ_{A}^{f}, λ_{A}^{f})$ is an intuitionistic fuzzy ideal of N, then A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of R.

Proof

Assume that $A^{f} = (μ_{A}^{f}, λ_{A}^{f})$ is an intuitionistic fuzzy ideal of N. Let i, x, y, z ∈ R. Then, there exist n, a, b, c ∈ N such that f(i) = n, f(x) = a, f(y) = b and f(z) = c. Then, (IF1) $μ_{A} (x - y) = μ_{A} (f (a) - f (b)) = μ_{A} (f (a - b)) = μ_{A}^{f} (a - b) \geq min {μ_{A}^{f} (a), μ_{A}^{f} (b)} = min {μ_{A} (f (a)), μ_{A} (f (b))} = min {μ_{A} (x), μ_{A} (y)}$ . Similarly, we can show that μ_A satisfies (IF2)–(IF6). Now, we consider (AF1) $λ_{A} (x - y) = λ_{A} (f (a) - f (b)) = λ_{A} (f (a - b)) = λ_{A}^{f} (a - b) \leq max {λ_{A}^{f} (a), λ_{A}^{f} (b)} = max {λ_{A} (f (a)), λ_{A} (f (b))} = max {λ_{A} (x), λ_{A} (y)}$ . Similarly, we can show that λ_A satisfies (AF2)–(AF6). Hence, A = (μ_A, λ_A) be an intuitionistic fuzzy ideal of R.

Let μ be a fuzzy set of a nonempty set X and $t \in [0, 1 - sup_{x \in X} μ (x)]$ . The mapping μ^T : X → [0, 1] is called a fuzzy translation [27] of μ if μ^T (x) = μ(x) + t, for all x ∈ X.

Theorem 8

Let N be a ternary near-ring, A = (μ_A, λ_A) be an intuitionistic fuzzy set of N, and $t \in [0, \frac{1}{2} (1 - sup_{x \in N} {μ_{A} (x) + λ_{A} (x)})]$ . Suppose that $μ_{A}^{T}$ and $λ_{A}^{T}$ are fuzzy translations of μ_A and λ_A with respect to t, respectively. Then, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N if and only if $A^{T} = (μ_{A}^{T}, λ_{A}^{T})$ is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. Let a ∈ N and $t = \frac{1}{2} (1 - sup_{x \in N} {μ_{A} (x) + λ_{A} (x)})$ . Then, $μ_{A}^{T} (a) + λ_{A}^{T} (a) = μ_{A} (a) + λ_{A} (a) + 2 t = μ_{A} (a) + λ_{A} (a) + 1 - sup_{x \in N} {μ_{A} (x) + λ_{A} (x)} \leq μ_{A} (a) + λ_{A} (a) + 1 - (μ_{A} (a) + λ_{A} (a)) = 1$ . Thus, $A^{T} = (μ_{A}^{T}, λ_{A}^{T})$ is an intuitionistic fuzzy set of N. Let i, x, y, z ∈ N. We have (IF1) $μ_{A}^{T} (x - y) = μ_{A} (x - y) + t \geq min {μ_{A} (x), λ_{A} (y)} + t = min {μ_{A} (x) + t, λ_{A} (y) + t} = min {μ_{A}^{T} (x), λ_{A}^{T} (y)}$ . It is not difficult to show that $μ_{A}^{T}$ satisfies (IF2)–(IF6). In addition, (AF1) $λ_{A}^{T} (x - y) = λ_{A} (x - y) + t \leq max {λ_{A} (x), λ_{A} (y)} + t = max {λ_{A} (x) + t, λ_{A} (y) + t} = max {λ_{A}^{T} (x), λ_{A}^{T} (y)}$ . Similarly, $λ_{A}^{T}$ satisfies (AF2)–(AF6). Hence, $A^{T} = (μ_{A}^{T}, λ_{A}^{T})$ is an intuitionistic fuzzy ideal of N.

Conversely, assume that $A^{T} = (μ_{A}^{T}, λ_{A}^{T})$ is an intuitionistic fuzzy ideal of N. Let i, x, y, z ∈ N. Then, (IF1) $μ_{A} (x - y) + t = μ_{A}^{T} (x - y) \geq min {μ_{A}^{T} (x), μ_{A}^{T} (y)} = min {μ_{A} (x), μ_{A} (y)} + t$ , implies that μ_A(x−y) ≥ min{μ_A(x), μ_A(y)} because t ≥ 0. Similarly, we can prove that μ_A satisfies (IF2)–(IF6). Next, we have (AF1) $λ_{A} (x - y) + t = λ_{A}^{T} (x - y) \leq max {λ_{A}^{T} (x), λ_{A}^{T} (y)} = max {λ_{A} (x), λ_{A} (y)}$ , and so λ_A(x − y) ≤ max{λ_A(x), λ_A(y)}, as t ≥ 0. Similarly, λ_A satisfies (AF2)–(AF6). Therefore, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N.

Let μ be a fuzzy set of a nonempty set X and m ∈ [0, 1]. The mapping μ^M : X → [0, 1] is called a fuzzy multiplication [27] of μ if μ^M(x) = mμ(x), for all x ∈ X.

Theorem 9

Let N be a ternary near-ring, A = (μ_A, λ_A) be an intuitionistic fuzzy set of N, and m ∈ (0, 1]. Suppose that $μ_{A}^{M}$ and $λ_{A}^{M}$ are fuzzy multiplications of μ_A and λ_A, respectively, where $μ_{A}^{M} (x) = {m μ}_{A} (x)$ and $λ_{A}^{M} (x) = {m λ}_{A} (x)$ , for all x ∈ N. Then, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N if and only if $A^{M} = (μ_{A}^{M}, λ_{A}^{M})$ is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. Obviously, $A^{M} = (μ_{A}^{M}, λ_{A}^{M})$ is an intuitionistic fuzzy set of N. Let i, x, y, z ∈ N. Then, (IF1) $μ_{A}^{M} (x - y) = {m μ}_{A} (x - y) \geq m min {μ_{A} (x), μ_{A} (y)} = min {{m μ}_{A} (x), {m μ}_{A} (y)} = min {μ_{A}^{M} (x), μ_{A}^{M} (y)}$ . The proofs of (IF2)–(IF6) are similar to that of (IF1). Next, we have $λ_{A}^{M} (x - y) = {m λ}_{A} (x - y) \leq m max {λ_{A} (x), λ_{A} (y)} = max {{m λ}_{A} (x), {m λ}_{A} (y)} = max {λ_{A}^{M} (x), λ_{A}^{M} (y)}$ . In the same way, we have $λ_{A}^{M}$ satisfying (AF2)–(AF6). Hence, $A^{M} = (μ_{A}^{M}, λ_{A}^{M})$ is an intuitionistic fuzzy ideal of N.

Conversely, assume that $A^{M} = (μ_{A}^{M}, λ_{A}^{M})$ is an intuitionistic fuzzy ideal of N. Let i, x, y, z ∈ N. Then, (IF1) ${m μ}_{A} (x - y) = μ_{A}^{M} (x - y) \geq min {μ_{A}^{M} (x), μ_{A}^{M} (y)} = m min {μ_{A} (x), μ_{A} (y)}$ . Because m > 0, μ_A(x − y) ≥ min{μ_A(x), μ_A(y)}. Uniformly, we have that μ_A satisfies (IF2)–(IF6). Now, consider (AF1) ${m λ}_{A} (x - y) = λ_{A}^{M} (x - y) \leq max {λ_{A}^{M} (x), λ_{A}^{M} (y)} = m max {λ_{A} (x), λ_{A} (y)}$ . This implies that

λ_{A} (x - y) \leq max {λ_{A} (x), λ_{A} (y)},

as m > 0. Similarly, λ_A satisfies (AF2)–(AF6). Therefore, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N.

Let μ be a fuzzy set of a nonempty set X, m ∈ [0, 1], and $t \in [0, 1 - sup_{x \in X} μ (x)]$ . The mapping μ^MT : X → [0, 1] is called a fuzzy magnified translation [28] of μ if μ^MT(x) = mμ(x)+t, for all x ∈ X. The following theorem immediately follows from Theorem 8 and Theorem 9.

Theorem 10

Let N be a ternary near-ring, A = (μ_A, λ_A) be an intuitionistic fuzzy set of N, $t \in [0, \frac{1}{2} (1 - sup_{x \in N} {μ_{A} (x) + λ_{A} (x)})]$ , and m ∈ (0, 1]. Suppose that μ^MT and λ^MT are fuzzy magnified translations of μ_A and λ_A with respect to t and m, respectively. Then, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N if and only if $A^{M T} = (μ_{A}^{M T}, λ_{A}^{M T})$ is an intuitionistic fuzzy ideal of N.

4. Noetherian and Artinian Ternary Near-Rings

In this section, we define the notions of Noetherian and Artinian ternary near-rings and characterize Northerian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.

Definition 8

A ternary near-ring N is called Noetherian (resp. Artinian) if N satisfies the ascending (resp. descending) chain condition on ideals of N, that is, for any ideals I₁, I₂, I₃, . . . of N, with

\begin{array}{l} I_{1} \subseteq I_{2} \subseteq I_{3} \subseteq \dots \subseteq I_{i} \subseteq \dots \\ (I_{1} \supseteq I_{2} \supseteq I_{3} \supseteq \dots \supseteq I_{i} \supseteq \dots), \end{array}

there exists n ∈ ℕ such that I_i = I_i₊₁ for all i ≥ n.

Theorem 11

If every intuitionistic fuzzy ideal of a ternary near-ring N has a finite image of values, then N is Noetherian.

Proof

Assume that every intuitionistic fuzzy ideal of a ternary near-ring N has the finite image of values. Suppose that N is not Noetherian, so there exists an ascending chain condition on ideals of N, that is, I₀ ⊆ I₁ ⊆ I₂ ⊆ · · ·. We define the intuitionistic fuzzy set A = (μ_A, λ_A) of N by

\begin{array}{l} μ_{A} (x) = {\begin{array}{l} \frac{1}{n + 1} & if x \in I_{n + 1} - I_{n}; \\ 1 & if x \in I_{0}; \\ 0 & if x \in S - ⋃_{n = 0}^{\infty} I_{n}, \end{array} \\ λ_{A} (x) = {\begin{array}{l} \frac{n}{n + 1} & if x \in I_{n + 1} - I_{n}; \\ 1 & if x \in I_{0}; \\ 0 & if x \in S - ⋃_{n = 0}^{\infty} I_{n}, \end{array} \end{array}

for all x ∈ N. It is not difficult to show that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. We have a contradiction because I₀ ⊆ I₁ ⊆ I₂ ⊆ ·· · is an infinitely ascending chain of ideals of N.

The proof of the following theorem is similar to that of Theorem 11.

Theorem 12

If every intuitionistic fuzzy ideal of a ternary near-ring N has a finite image of values, then N is Artinian.

Theorem 13

A ternary near-ring N is Noetherian if and only if the set of values of any intuitionistic fuzzy ideal of N is a well-ordered subset of [0, 1].

Proof

Assume that N is Noetherian. Suppose that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N, which is not a well-ordered subset of [0, 1]. Then, there exists an infinite descending sequence ${t_{n}}_{n = 1}^{\infty}$ such that μ_A(x) = t_n and λ_A(x) ≤ 1 − t_n, for some x ∈ N. We define I_n = {x ∈ N | μ_A(x) ≥ t_n} and J_n = {x ∈ N | λ_A(x) ≤ 1−t_n}. By Theorem 1, I_n and J_n are ideals of N, for all n ∈ ℕ. It follows that I₁ ⊂ I₂ ⊂ I₃ ⊂ ·· · and J₁ ⊂ J₂ ⊂ J₃ ⊂ ·· · are strictly infinite ascending chains of ideals of N. This contradicts our hypothesis.

Conversely, suppose that N is not Noetherian. Then, there exists a strictly infinite ascending chain of ideals of N, namely, I₁ ⊆ I₂ ⊆ I₃ ⊆ ·· ·. Let $I = ⋃_{n \in ℕ} I_{n}$ . It easy to show that I is ideal for N. Next, we define the intuitionistic fuzzy set of A = (μ_A, λ_A) of N by

\begin{array}{l} μ_{A} (x) = {\begin{array}{l} \frac{1}{m} & if m = min {n \in ℕ | x \in I_{n}}; \\ 0 & if x \notin I; \end{array} \\ λ_{A} (x) = {\begin{array}{l} \frac{m - 1}{m + 1} & if m = min {n \in ℕ | x \in I_{n}}; \\ 1 & if x \notin I, \end{array} \end{array}

for all x ∈ N. Obviously, A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N. Because the chain is not finite, A = (μ_A, λ_A) has an infinite ascending sequence of values. This is a contradiction with the idea that the set of values of the intuitionistic fuzzy ideal is not a well-ordered subset of [0, 1].

Theorem 14

A ternary near-ring N is both Noetherian and Artinian if and only if every intuitionistic fuzzy ideal of N has a finite image of values.

Proof

Suppose that A = (μ_A, λ_A) is an intuitionistic fuzzy ideal of N such that Im(μ_A) and Im(λ_A) are infinite. By Theorem 1, U(μ_A; t_n) and L(λ_A; s_m) are ideals of N, for all m, n ∈ ℕ. Because N is Noetherian and by Theorem 13, Im(μ_A) and Im(λ_A) are well-ordered subsets of [0, 1]. Then, we can separate them into two cases, as follows.

Case 1

Let t₁ < t₂ < t₃ < · · · be an increasing sequence in Im(μ_A) and s₁ > s₂ > s₃ > · · · be a decreasing sequence in Im(λ_A). It follows that U(μ_A; t₁) ⊃ U(μ_A; t₂) ⊃ U(μ_A; t₃) ⊃ ·· · and L(λ_A; s₁) ⊃ L(λ_A; s₂) ⊃ L(λ_A; s₃) ⊃ · · · are exactly descending chains of ideals of N. Because N is Artinian, there exist i, j ∈ ℕ such that U(μ_A; t_i) = U(μ_A; t_i₊_k) and L(λ_A; s_j) = L(λ_A; s_j₊_l), for all k, l ∈ ℕ. It turns out that t_i = t_i₊_k and s_j = s_j₊_l, for all k, l ∈ ℕ. This is a contradiction.

Case 2

Let t₁ > t₂ > t₃ > · · · be a decreasing sequence. in Im(μ_A) and s₁ < s₂ < s₃ < · · · be an increasing sequence in Im(λ_A). It follows that U(μ_A; t₁) ⊂ U(μ_A; t₂) ⊂ U(μ_A; t₃) ⊂ ·· · and L(λ_A; s₁) ⊂ L(λ_A; s₂) ⊂ L(λ_A; s₃) ⊂ ·· · are absolutely ascending chains of ideals of N. Because N is Noetherian, there exist i, j ∈ ℕ such that U(μ_A; t_i) = U(μ_A; t_i₊_k) and L(λ_A; s_j) = L(λ_A; s_j₊_l), for all k, l ∈ ℕ. It follows that t_i = t_i₊_k and s_j = s_j₊_l, for all k, l ∈ ℕ. We have a contradiction.

Conversely, it follows by Theorem 11 and Theorem 12.

5. Conclusion: We introduced the concept of the intuitionistic fuzzy ideal in ternary near-rings as a generalization of their fuzzy ideals and studied some of their properties. We also presented the notions of Noetherian and Artinian ternary near-rings and characterized some of their properties using their intuitionistic fuzzy ideals. In the future, we would like to investigate some of the basic properties of the concepts of fuzzy quasi-ideals and fuzzy bi-ideals in ternary near-rings. Next, we will study the concepts of intuitionistic fuzzy quasi-ideals and intuitionistic fuzzy bi-ideals in ternary near-rings as generalizations of their fuzzy quasi-ideals and fuzzy bi-ideals, respectively.

Acknowledgments: This research was financially supported by the Faculty of Science, Mahasarakham University.

Conflict of Interest: No potential conflict of interest relevant to this article is reported.

TABLES

+	a	b	c	d
a	a	b	c	d
b	b	a	d	c
c	c	d	b	a
d	d	c	a	b

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Biography

Warud Nakkhasen received the Ph.D. in Mathematics from Khon Kaen University, Thailand in 2019. Currently, he is a lecturer at the Department of Mathematics, Faculty of Science, Mahasarakham University, Thailand. He research fields focus on algebraic structure, algebraic hyperstructure, fuzzy set theory, and intuitionistic fuzzy set theory.

E-mail: warud.n@msu.ac.th

: December 2020, 20 (4)

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Mahasarakham University
10.13039/501100007288