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.
In this section, we present the basic definitions that are used in the following sections of this paper.
A ternary semi-group is an algebraic structure (N,+, [ ]) such that N is a nonempty set, and [ ] : N^{3} → 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.
Let A, B, and C be nonempty subsets of a ternary near-ring N. Then, [ABC] = {[abc] ∈ N | a ∈ A, b ∈ B, c ∈ C}.
Let N be a nonempty set together with a binary operation + and a ternary operation [ ] : N^{3} → 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.”
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.
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.
Let N = {a, b, c, d} be a set with a binary operation + on N as follows:
+ | a | b | c | d |
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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.
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
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}.
In this section, we introduce the concept of intuitionistic fuzzy ideals of ternary near-rings and investigate some of their properties.
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).
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.
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.
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).
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.
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 t_{1} ∈ [0, 1] such that t_{1} = min{μ_{A}(x), μ_{A}(y)}. Then, μ_{A}(x) ≥ t_{1} and μ_{A}(y) ≥ t_{1}, so x, y ∈ U(μ_{A}; t_{1}). By assumption, x−y ∈ U(μ_{A}; t_{1}). Thus, μ_{A}(x − y) ≥ t_{1} = min{μ_{A}(x), μ_{A}(y)}. (IF2) Let t_{2} ∈ [0, 1] such that t_{2} = min{μ_{A}(x), μ_{A}(y), μ_{A}(z)}. Then, μ_{A}(x) ≥ t_{2}, μ_{A}(y) ≥ t_{2}, μ_{A}(z) ≥ t_{2}, that is, x, y, z ∈ U(μ_{A}; t_{2}). By assumption, [xyz] ∈ U(μ_{A}; t_{2}). Thus, μ_{A}([xyz]) ≥ t_{2} = min{μ_{A}(x), μ_{A}(y), μ_{A}(z)}. (IF3) Let t_{3} ∈ [0, 1] such that t_{3} = μ_{A}(x). Then x ∈ U(μ_{A}; t_{3}). So, y + x − y ∈ U(μ_{A}; t_{3}). Hence, μ_{A}(y + x − y) ≥ t_{3} = μ_{A}(x). (IF4) Let t_{4} ∈ [0, 1] such that t_{4} = μ_{A}(z). Then z ∈ U(μ_{A}; t_{4}). Thus, [xyz] ∈ U(μ_{A}; t_{4}), and then μ_{A}([xyz]) ≥ t_{4} = μ_{A}(z). (IF5) Let t_{5} ∈ [0, 1] such that t_{5} = μ_{A}(i). Then i ∈ U(μ_{A}; t_{5}). Thus, [(x + i)yz] − [xyz] ∈ U(μ_{A}; t_{5}), that is, μ_{A}([(x + i)yz] − [xyz]) ≥ t_{5} = μ_{A}(i). (IF6) Let t_{6} ∈ [0, 1] such that t_{6} = μ_{A}(i). Then, i ∈ U(μ_{A}; t_{6}). Thus, [x(y+i)z]− [xyz] ∈ U(μ_{A}; t_{6}). It follows that μ_{A}([x(y + i)z] − [xyz]) ≥ t_{6} = μ_{A}(i). Therefore, μ_{A} satisfies (IF1)–(IF6). Similarly, we can prove that λ_{A} satisfies (AF1)–(AF6). This completes the proof.
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.
Let N be a ternary near-ring. Then, A = (μ_{A}, λ_{A}) is an intuitionistic fuzzy ideal of N if and only if
Assume that A = (μ_{A}, λ_{A}) is an intuitionistic fuzzy ideal of N. Let i, x, y, z ∈ N. (IF1)
Conversely, assume that
The following theorem immediately follows from Theorem 2.
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
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.
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,
The proofs of (IF2)–(IF6) are similar to that of (IF1). Next, (AF1) we have
The proofs of (AF2)–(AF6) are similar to that of (AF1). Therefore, A ∪ B is an intuitionistic fuzzy ideal of N.
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.
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
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
Assume that A = (μ_{A}, λ_{A}) is an intuitionistic fuzzy ideal of R. Let i, x, y, z ∈ N. Then, (IF1)
If we strengthen the condition of f, we can construct the converse of Theorem 6 as follows.
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
Assume that
Let μ be a fuzzy set of a nonempty set X and
Let N be a ternary near-ring, A = (μ_{A}, λ_{A}) be an intuitionistic fuzzy set of N, and
Assume that A = (μ_{A}, λ_{A}) is an intuitionistic fuzzy ideal of N. Let a ∈ N and
Conversely, assume that
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.
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
Assume that A = (μ_{A}, λ_{A}) is an intuitionistic fuzzy ideal of N. Obviously,
Conversely, assume that
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
Let N be a ternary near-ring, A = (μ_{A}, λ_{A}) be an intuitionistic fuzzy set of N,
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.
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 I_{1}, I_{2}, I_{3}, . . . of N, with
there exists n ∈ ℕ such that I_{i} = I_{i}_{+1} for all i ≥ n.
If every intuitionistic fuzzy ideal of a ternary near-ring N has a finite image of values, then N is Noetherian.
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, I_{0} ⊆ I_{1} ⊆ I_{2} ⊆ · · ·. We define the intuitionistic fuzzy set A = (μ_{A}, λ_{A}) of N by
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 I_{0} ⊆ I_{1} ⊆ I_{2} ⊆ ·· · is an infinitely ascending chain of ideals of N.
The proof of the following theorem is similar to that of Theorem 11.
If every intuitionistic fuzzy ideal of a ternary near-ring N has a finite image of values, then N is Artinian.
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].
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
Conversely, suppose that N is not Noetherian. Then, there exists a strictly infinite ascending chain of ideals of N, namely, I_{1} ⊆ I_{2} ⊆ I_{3} ⊆ ·· ·. Let
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].
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.
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.
Let t_{1} < t_{2} < t_{3} < · · · be an increasing sequence in Im(μ_{A}) and s_{1} > s_{2} > s_{3} > · · · be a decreasing sequence in Im(λ_{A}). It follows that U(μ_{A}; t_{1}) ⊃ U(μ_{A}; t_{2}) ⊃ U(μ_{A}; t_{3}) ⊃ ·· · and L(λ_{A}; s_{1}) ⊃ L(λ_{A}; s_{2}) ⊃ L(λ_{A}; s_{3}) ⊃ · · · 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.
Let t_{1} > t_{2} > t_{3} > · · · be a decreasing sequence. in Im(μ_{A}) and s_{1} < s_{2} < s_{3} < · · · be an increasing sequence in Im(λ_{A}). It follows that U(μ_{A}; t_{1}) ⊂ U(μ_{A}; t_{2}) ⊂ U(μ_{A}; t_{3}) ⊂ ·· · and L(λ_{A}; s_{1}) ⊂ L(λ_{A}; s_{2}) ⊂ L(λ_{A}; s_{3}) ⊂ ·· · 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.
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.
This research was financially supported by the Faculty of Science, Mahasarakham University.
No potential conflict of interest relevant to this article is reported.
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E-mail: warud.n@msu.ac.th