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Intuitionistic Fuzzy Ideals of Ternary Near-Rings
International Journal of Fuzzy Logic and Intelligent Systems 2020;20(4):290-297
Published online December 25, 2020
© 2020 Korean Institute of Intelligent Systems.

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 [46] 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 [810].

The concept of intuitionistic fuzzy sets was introduced by Atanassov [1113] 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 [1419]. 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 [ ] : N3N is a ternary operation satisfying the following associative law: [[abc]de] = [a[bcd]e] = [ab[cde]], for all a, b, c, d, eN.

Definition 2 [26]

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

Definition 3 [23]

Let N be a nonempty set together with a binary operation + and a ternary operation [ ] : N3N. 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, dN.

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 abT and [abc] ∈ T, for all a, b, cT.

Definition 5 [24]

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

  • (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:

+abcd
aabcd
bbadc
ccdba
ddcab

The ternary operation [ ] on N is defined by [xyz] = z for all x, y, zN. 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 : NR 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, cN.

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) = {xX | μ(x) ≥ t} is called an upper-level set of μ, and the set L(μ; t) = {xX | μ(x) ≤ t} is called a lower-level set of μ, where t ∈ [0, 1]. The complement of μ denoted by μc is the fuzzy set of X defined by μc(x) = 1 − μ(x), for all xX. The intersection and union of two fuzzy sets μ and λ of X, denoted by μλ and μλ, respectively, are defined by letting xX, (μλ)(x) = min{μ(x), λ(x)} and (μλ)(x) = max{μ(x), λ(x)}, respectively.

The concept of intuitionistic fuzzy sets was introduced by Atanassov [1113] 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,μA(x),λA(x))|xX},

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

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 = (μA, λA) of a ternary near-ring (N,+, [ ]) is called an intuitionistic fuzzy ideal of N if it satisfies for every i, x, y, zN,

  • (IF1) μA(xy) ≥ min{μA(x), μA(y)};

  • (IF2) μA([xyz]) ≥ min{μA(x), μA(y), μA(z)};

  • (IF3) μA(y + xy) ≥ μA(x);

  • (IF4) μA([xyz]) ≥ μA(z);

  • (IF5) μA([(x + i)yz] − [xyz]) ≥ μA(i);

  • (IF6) μA([x(y + i)z] − [xyz]) ≥ μA(i);

  • (AF1) λA(xy) ≤ max{λA(x), λA(y)};

  • (AF2) λA([xyz]) ≤ max{λA(x), λA(y), λA(z)};

  • (AF3) λA(y + xy) ≤ λA(x);

  • (AF4) λA([xyz]) ≤ λA(z);

  • (AF5) λA([(x + i)yz] − [xyz]) ≤ λA(i);

  • (AF6) λA([x(y + i)z] − [xyz]) ≤ λA(i).

Example 2

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

Proposition 1

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

Proof

Assume that A = (μA, λA) is an intuitionistic fuzzy ideal of a ternary near-ring N. Let xN. Then, μA(0) = μA(xx) ≥ min{μA(x), μA(x)} = μA(x) and λA(0) = λA(xx) ≤ max{λA(x), λA(x)} = λA(x).

Theorem 1

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

Proof

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

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

Example 3

In Example 2, it is clear that A = (μA, λA) is an intuitionistic fuzzy ideal of N. Then, we can show that the upper-level sets of N are U(μA; μA(a)) = {a}, U(μA; μA(b)) = {a, b}, U(μA; μA(c)) = N, and U(μA; μA(d)) = N. Also, the lower-level sets of N are L(λA; λA(a)) = {a}, L(λA; λA(b)) = {a, b}, L(λA; λA(c)) = N, and L(λA; λA(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 = (μA, λA) is an intuitionistic fuzzy ideal of N if and only if Ac=(λAc,μAc) is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μA, λA) is an intuitionistic fuzzy ideal of N. Let i, x, y, zN. (IF1) λAc(x-y)=1-λA(x-y)1-max{λA(x),λA(y)}=min{1-λA(x),1-λA(y)}=min{λAc(x),λAc(y)}. Similarly, we can show that λAc satisfies (IF2)–(IF6). Next, we consider (AF1) μAc(x-y)=1-μA(x-y)1-min{μA(x),μA(y)}=max{1-μA(x),1-μA(y)}=max{μAc(x),μAc(y)}. Similarly, we can show that (AF2)–(AF6). Hence, Ac=(λAc,μAc) is an intuitionistic fuzzy ideal of N.

Conversely, assume that Ac=(λAc,μAc) is an intuitionistic fuzzy ideal of N. Let i, x, y, zN. (IF1) 1-μA(x-y)=μAc(x-y)max{μAc(x),μAc(y)}=max{1-μA(x),1-μA(y)}=1-min{μA(x),μA(y)}. This implies that μA(xy) ≥ min{μA(x), μA(y)}. Similarly, we can show that μA satisfies (IF2)–(IF6). Next, we consider (AF1) 1-λA(x-y)=λAc(x-y)min{λAc(x),λAc(y)}=min{1-λA(x),1-λA(y)}=1-max{λA(x),λA(y)}. It follows that λA(xy) ≤ max{λA(x), λA(y)}. Similarly, we can show that λA satisfies (AF2)–(AF6). Therefore, A = (μA, λA) is an intuitionistic fuzzy ideal of N.

The following theorem immediately follows from Theorem 2.

Theorem 3

Let A = (μA, λA) be an intuitionistic fuzzy set of a ternary near-ring N. Then, A = (μA, λA) is an intuitionistic fuzzy ideal of N if and only if Aμ=(μA,μAc) and Aλ=(λAc,λA) are intuitionistic fuzzy ideals of N.

Theorem 4

If A = (μA, λA) and B = (μB, λB) are intuitionistic fuzzy ideals of a ternary near-ring N, then AB = (μAμB, λAλB) is also an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μA, λA) and B = (μB, λB) are intuitionistic fuzzy ideals of a ternary near-ring N. Clearly, AB is an intuitionistic fuzzy set of N. Let i, x, y, zN. (IF1) Then,

(μAμB)(x-y)=max{μA(x-y),μA(x-y)}max{min{μA(x),μA(y)},min{μB(x),μB(y)}}=min{max{μA(x),μB(x)},max{μA(y),μB(y)}}=min{(μAμB)(x),(μAμB)(y)}.

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

(λAλB)(x-y)=min{λA(x-y),λB(x-y)}min{max{λA(x),λA(y)},max{λB(x),λB(y)}}=max{min{λA(x),λB(x)},min{λA(y),λB(y)}}=max{(λAλB)(x),(λAλB)(y)}.

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

Theorem 5

If A = (μA, λA) and B = (μB, λB) are intuitionistic fuzzy ideals of a ternary near-ring N, then AB = (μAμB, λAλB) is also an intuitionistic fuzzy ideal of N.

Proof

The proof is similar to Theorem 4.

Let f : NR be a homomorphism of ternary near-rings. For any A = (μA, λA) of R, we define a new Af=(μAf,λAf) of N by μAf(x)=μA(f(x)) and λAf(x)=λA(f(x)), for all xN.

Theorem 6

Let f : NR be a homomorphism of ternary near-rings. If A = (μA, λA) is an intuitionistic fuzzy ideal of R, then Af=(μAf,λAf) is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μA, λA) is an intuitionistic fuzzy ideal of R. Let i, x, y, zN. Then, (IF1) μAf(x-y)=μA(f(x-y))=μA(f(x)-f(y))min{μA(f(x)),μA(f(y))}=min{μAf(x),μAf(y)}. Similarly, we can prove that μAf satisfies (IF2)–(IF6). Now, (AF1) λAf(x-y)=λA(f(x-y))=λA(f(x)-f(y))max{λA(f(x)),λA(f(y))}=max{λAf(x),λAf(y)}. Similarly, we can prove that λAf satisfies (AF2)–(AF6). Therefore, Af=(μAf,λAf) 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 : NR be an epimorphism of ternary near-rings, and let A = (μA, λA) be an intuitionistic fuzzy set of R. If Af=(μAf,λAf) is an intuitionistic fuzzy ideal of N, then A = (μA, λA) is an intuitionistic fuzzy ideal of R.

Proof

Assume that Af=(μAf,λAf) is an intuitionistic fuzzy ideal of N. Let i, x, y, zR. Then, there exist n, a, b, cN such that f(i) = n, f(x) = a, f(y) = b and f(z) = c. Then, (IF1) μA(x-y)=μA(f(a)-f(b))=μA(f(a-b))=μAf(a-b)min{μAf(a),μAf(b)}=min{μA(f(a)),μA(f(b))}=min{μA(x),μA(y)}. Similarly, we can show that μA satisfies (IF2)–(IF6). Now, we consider (AF1) λA(x-y)=λA(f(a)-f(b))=λA(f(a-b))=λAf(a-b)max{λAf(a),λAf(b)}=max{λA(f(a)),λA(f(b))}=max{λA(x),λA(y)}. Similarly, we can show that λA satisfies (AF2)–(AF6). Hence, A = (μA, λA) be an intuitionistic fuzzy ideal of R.

Let μ be a fuzzy set of a nonempty set X and t[0,1-supxXμ(x)]. The mapping μT : X → [0, 1] is called a fuzzy translation [27] of μ if μT (x) = μ(x) + t, for all xX.

Theorem 8

Let N be a ternary near-ring, A = (μA, λA) be an intuitionistic fuzzy set of N, and t[0,12(1-supxN{μA(x)+λA(x)})]. Suppose that μAT and λAT are fuzzy translations of μA and λA with respect to t, respectively. Then, A = (μA, λA) is an intuitionistic fuzzy ideal of N if and only if AT=(μAT,λAT) is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μA, λA) is an intuitionistic fuzzy ideal of N. Let aN and t=12(1-supxN{μA(x)+λA(x)}). Then, μAT(a)+λAT(a)=μA(a)+λA(a)+2t=μA(a)+λA(a)+1-supxN{μA(x)+λA(x)}μA(a)+λA(a)+1-(μA(a)+λA(a))=1. Thus, AT=(μAT,λAT) is an intuitionistic fuzzy set of N. Let i, x, y, zN. We have (IF1) μAT(x-y)=μA(x-y)+tmin{μA(x),λA(y)}+t=min{μA(x)+t,λA(y)+t}=min{μAT(x),λAT(y)}. It is not difficult to show that μAT satisfies (IF2)–(IF6). In addition, (AF1) λAT(x-y)=λA(x-y)+tmax{λA(x),λA(y)}+t=max{λA(x)+t,λA(y)+t}=max{λAT(x),λAT(y)}. Similarly, λAT satisfies (AF2)–(AF6). Hence, AT=(μAT,λAT) is an intuitionistic fuzzy ideal of N.

Conversely, assume that AT=(μAT,λAT) is an intuitionistic fuzzy ideal of N. Let i, x, y, zN. Then, (IF1) μA(x-y)+t=μAT(x-y)min{μAT(x),μAT(y)}=min{μA(x),μA(y)}+t, implies that μA(xy) ≥ min{μA(x), μA(y)} because t ≥ 0. Similarly, we can prove that μA satisfies (IF2)–(IF6). Next, we have (AF1) λA(x-y)+t=λAT(x-y)max{λAT(x),λAT(y)}=max{λA(x),λA(y)}, and so λA(xy) ≤ max{λA(x), λA(y)}, as t ≥ 0. Similarly, λA satisfies (AF2)–(AF6). Therefore, A = (μA, λA) is an intuitionistic fuzzy ideal of N.

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

Theorem 9

Let N be a ternary near-ring, A = (μA, λA) be an intuitionistic fuzzy set of N, and m ∈ (0, 1]. Suppose that μAM and λAM are fuzzy multiplications of μA and λA, respectively, where μAM(x)=mμA(x) and λAM(x)=mλA(x), for all xN. Then, A = (μA, λA) is an intuitionistic fuzzy ideal of N if and only if AM=(μAM,λAM) is an intuitionistic fuzzy ideal of N.

Proof

Assume that A = (μA, λA) is an intuitionistic fuzzy ideal of N. Obviously, AM=(μAM,λAM) is an intuitionistic fuzzy set of N. Let i, x, y, zN. Then, (IF1) μAM(x-y)=mμA(x-y)mmin{μA(x),μA(y)}=min{mμA(x),mμA(y)}=min{μAM(x),μAM(y)}. The proofs of (IF2)–(IF6) are similar to that of (IF1). Next, we have λAM(x-y)=mλA(x-y)mmax{λA(x),λA(y)}=max{mλA(x),mλA(y)}=max{λAM(x),λAM(y)}. In the same way, we have λAM satisfying (AF2)–(AF6). Hence, AM=(μAM,λAM) is an intuitionistic fuzzy ideal of N.

Conversely, assume that AM=(μAM,λAM) is an intuitionistic fuzzy ideal of N. Let i, x, y, zN. Then, (IF1) mμA(x-y)=μAM(x-y)min{μAM(x),μAM(y)}=mmin{μA(x),μA(y)}. Because m > 0, μA(xy) ≥ min{μA(x), μA(y)}. Uniformly, we have that μA satisfies (IF2)–(IF6). Now, consider (AF1) mλA(x-y)=λAM(x-y)max{λAM(x),λAM(y)}=mmax{λA(x),λA(y)}. This implies that

λA(x-y)max{λA(x),λA(y)},

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

Let μ be a fuzzy set of a nonempty set X, m ∈ [0, 1], and t[0,1-supxXμ(x)]. The mapping μMT : X → [0, 1] is called a fuzzy magnified translation [28] of μ if μMT(x) = (x)+t, for all xX. The following theorem immediately follows from Theorem 8 and Theorem 9.

Theorem 10

Let N be a ternary near-ring, A = (μA, λA) be an intuitionistic fuzzy set of N, t[0,12(1-supxN{μA(x)+λA(x)})], and m ∈ (0, 1]. Suppose that μMT and λMT are fuzzy magnified translations of μA and λA with respect to t and m, respectively. Then, A = (μA, λA) is an intuitionistic fuzzy ideal of N if and only if AMT=(μAMT,λAMT) 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

I1I2I3Ii(I1I2I3Ii),

there exists n ∈ ℕ such that Ii = Ii+1 for all in.

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, I0I1I2 ⊆ · · ·. We define the intuitionistic fuzzy set A = (μA, λA) of N by

μA(x)={1n+1if xIn+1-In;1if xI0;0if xS-n=0In,λA(x)={nn+1if xIn+1-In;1if xI0;0if xS-n=0In,

for all xN. It is not difficult to show that A = (μA, λA) is an intuitionistic fuzzy ideal of N. We have a contradiction because I0I1I2 ⊆ ·· · 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 = (μA, λA) is an intuitionistic fuzzy ideal of N, which is not a well-ordered subset of [0, 1]. Then, there exists an infinite descending sequence {tn}n=1 such that μA(x) = tn and λA(x) ≤ 1 − tn, for some xN. We define In = {xN | μA(x) ≥ tn} and Jn = {xN | λA(x) ≤ 1−tn}. By Theorem 1, In and Jn are ideals of N, for all n ∈ ℕ. It follows that I1I2I3 ⊂ ·· · and J1J2J3 ⊂ ·· · 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, I1I2I3 ⊆ ·· ·. Let I=nIn. It easy to show that I is ideal for N. Next, we define the intuitionistic fuzzy set of A = (μA, λA) of N by

μA(x)={1mif m=min{n|xIn};0if xI;λA(x)={m-1m+1if m=min{n|xIn};1if xI,

for all xN. Obviously, A = (μA, λA) is an intuitionistic fuzzy ideal of N. Because the chain is not finite, A = (μA, λA) 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 = (μA, λA) is an intuitionistic fuzzy ideal of N such that Im(μA) and Im(λA) are infinite. By Theorem 1, U(μA; tn) and L(λA; sm) are ideals of N, for all m, n ∈ ℕ. Because N is Noetherian and by Theorem 13, Im(μA) and Im(λA) 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(μA) and s1 > s2 > s3 > · · · be a decreasing sequence in Im(λA). It follows that U(μA; t1) ⊃ U(μA; t2) ⊃ U(μA; t3) ⊃ ·· · and L(λA; s1) ⊃ L(λA; s2) ⊃ L(λA; s3) ⊃ · · · are exactly descending chains of ideals of N. Because N is Artinian, there exist i, j ∈ ℕ such that U(μA; ti) = U(μA; ti+k) and L(λA; sj) = L(λA; sj+l), for all k, l ∈ ℕ. It turns out that ti = ti+k and sj = sj+l, for all k, l ∈ ℕ. This is a contradiction.

Case 2

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


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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




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