International Mathematical Forum, Vol. 7, 2012, no. 41, 2045 - 2059

L-Fuzzy Ternary Subnear-Rings1

Warud Nakkhasen2 and Bundit Pibaljommee3

Department of , Faculty of Science Khon Kaen University, Khon Kaen 40002, Thailand Centre of Excellence in Mathematics CHE, Si Ayuttaya Rd. Bangkok 10400, Thailand [email protected]

Abstract

In this article, we give the concept of a ternary near-, a ternary subnear-ring and an ideal of a ternary near-ring. Then we investigate some properties of an L-fuzzy ternary subnear-ring and an L-fuzzy ideal of a ternary near-ring, where L is a complete with the greatest element 1 and the least element 0. Moreover, we show that the lattice of all normal L-fuzzy ternary subnear-rings is a complete sublattice of the lattice of all L-fuzzy ternary subnear-rings, where L is a complemented .

Mathematics Subject Classification: 03E72, 06D72, 16Y30, 20N10

Keywords: ternary near-ring, L-fuzzy ternary subnear-ring, L-fuzzy ideal, normal L-fuzzy subset

1 Introduction

The notion of fuzzy set was introduced first by Zadeh ([23]) as a function from set X to the unit interval [0, 1]. The first inspiration application to many alge- braic structures was the concept of fuzzy introduced by Rosenfeld ([20]). Liu ([18]) has studied fuzzy ideals of a ring, and many researchers, e.g., [3], [13], [14], [22], extended the concepts. The notion of fuzzy subnear-ring, fuzzy left (resp. right) ideals in near-ring was introduced by Abou-Zaid in ([1]). They 1This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. 2e-mail: warut [email protected] 3Corresponding Author. e-mail: [email protected] 2046 W. Nakkhasen and B. Pibaljommee have been studied by many authors, see ([6], [9], [12]). The ternary algebraic system so-called triplexes has been introduced first by Lehmer ([16]) in 1932. This investigation was certain to commutative ternary groups. Dutta and Kar ([4]) introduced the notion of ternary which is a generalization of the ternary ring introduced by Lister ([17]). The notion of fuzzy ideals and fuzzy quasi-ideals in ternary was investigated first by Kavikumar and Khamis ([10]) in 2007 and this concept was applied to many concepts, e.g., fuzzy bi-deals in ternary semirings by Kavikumar, Khamis and Jun, ([11]) in 2009, L-fuzzy ternary semiring and k-fuzzy ideal of ternary semirings by Chin- ram and Malee ([2], [19]) in 2010. In this work, we give the notion of ternary near-rings, ideals of ternary near-rings, L-fuzzy subnear-rings, L-fuzzy ideals of ternary near-rings and investigate some properties of L-fuzzy subnear-rings and L-fuzzy ideals of ternary near-rings. Moreover, using the concept of nor- mal L-fuzzy ideals in semirings introduced by Jun, Neggers and Kim ([7]), we consider normal L-fuzzy ternary subnear-rings of a ternary near-ring, where L is a complemented distributive lattice.

2 Preliminaries

In this section, we review some definitions and some results which will be used in later sections.

Definition 2.1 ([12]) A nonempty set N together with binary operations + and · is called a near-ring if

(i) (N,+) is a group,

(ii) (N,·) is a and

(iii) x · (y + z)=x · y + x · z, for all x, y, z ∈ N.

Now we mention the concept of ternary semigroup and left ideal of ternary semigroup defined in ([21]).

Definition 2.2 ([21]) A ternary semigroup is an (S, f) such that S is a nonempty set and f : S3 → S is a ternary operation satisfying the following associative law:

f(f(a, b, c),d,e)=f(a, f(b, c, d),e)=f(a, b, f(c, d, e)), for all a, b, c, d, e ∈ S.

Definition 2.3 ([21]) A nonempty subset I ⊆ S is called an ideal of a ternary semigroup (S, f) if f(S, S, I) ⊆ I, f(I, S, S) ⊆ I, f(S, I, S) ⊆ I. L-fuzzy ternary subnear-rings 2047

Note that I is a left (resp. right, lateral) ideal of S if f(S, S, I) ⊆ I (resp. f(I, S, S) ⊆ I, f(S, I, S) ⊆ I). Throughout this paper, let L =(L, ≤, ∧, ∨)be a complete lattice which has the least and the greatest elements, say 0 and 1, respectively unless specified otherwise. Let X be a nonempty set. An L-fuzzy subset of X is a mapping μ : X → L. We denote by F (X) the set of all L-fuzzy subsets of X.Forμ, ν ∈ F (X), μ ⊆ ν means μ(x) ≤ ν(x) for all x ∈ X. It is easy to see that F (X)=(F (X), ⊆, ∧, ∨) is a complete lattice, which has the least and the greatest elements, say 0 and 1, respectively, where 0(x) = 0 and 1(x) = 1 for all x ∈ X. Let μ ∈ F (X). For t ∈ L, the set μt = {x ∈ X | μ(x) ≥ t} is called a level subset of the L-fuzzy subset μ.

Proposition 2.4 (Proposition 8 of [8]) Let f be a mapping from a set X to a set Y and μ ∈ F (X). Then for every t ∈ L, t =0 , (f(μ))t = f(μt−s). 0

We call ν the image of μ under f denoted by f(μ). Conversely, for ν ∈ F (f(X)), define μ ∈ F (X)byμ(x)=ν(f(x)) for all x ∈ X, and we call μ the preimage of ν under f denoted by f −1(ν).

3 Ternary near-rings

In this section, we apply the concept of ternary semiring defined in ([4]) to introduce the concepts of ternary near-rings, ternary subnear-rings and ideals of a ternary near-ring. Definition 3.1 A tri-tuple (N,+, ·) consisting of a nonempty set N, a bi- nary operation + on N and a ternary operation · on N is called a ternary near-ring if (i)(N,+) is a group,

(ii)(N,·) is a ternary semigroup and

(iii) ab(c + d)=abc + abd, for all a, b, c, d ∈ N.

Definition 3.2 A nonempty set T of N is called a ternary subnear-ring of N if (T,+) is a subgroup of (N,+) and t1t2t3 ∈ T for all t1,t2,t3 ∈ T . 2048 W. Nakkhasen and B. Pibaljommee

Definition 3.3 An ideal I of a ternary near-ring N is a nonempty subset of N such that

(i)(I,+) is a normal subgroup of (N,+),

(ii) NNI ⊆ I and

(iii)(x + i)yz − xyz ∈ I for any i ∈ I and any x, y, z ∈ N.

Example 3.4 Let N={a, b, c, d } be a set with a binary operation as follows:

+ abcd a abcd b badc c cdba d dcab

Define the ternary operation · : N 3 → N by xyz = z for all x, y, z ∈ N. Then we can easily see that (N,+, ·) is a ternary near-ring. Let I = {a, b}. Then (I,+, ·) is a ternary subnear-ring of (N,+, ·). Moreover, (I,+, ·) is an ideal of (N,+, ·).

We note that I is a left ideal of N if I satisfies (i) and (ii), and I is a right ideal of N if I satisfies (i) and (iii). We note that arbitrary intersection of ternary subnear-rings (resp. left ideals, right ideals, ideals) of a ternary near-ring is again a ternary subnear-ring (resp. left ideal, right ideal, ideal).

Definition 3.5 Let N and R be ternary near-rings. A mapping ϕ : N → R is called a homomorphism if ϕ(x+y)=ϕ(x)+ϕ(y) and ϕ(xyz)=ϕ(x)ϕ(y)ϕ(z) for all x, y, z ∈ N.

4 L-fuzzy ideals of ternary near-rings

In this section, we introduce the concepts of L-fuzzy ternary subnear-ring, L- fuzzy ideal, characteristic and L-fuzzy characteristic ternary subnear-ring of ternary near-ring and study some of their fundamental properties.

Definition 4.1 Let N be a ternary near-ring and μ be a fuzzy subset of N. We say that μ is an L-fuzzy ternary subnear-ring of N if

(i) μ(x − y) ≥ inf{μ(x),μ(y)} and

(ii) μ(xyz) ≥ inf{μ(x),μ(y),μ(z)}, for all x, y, z ∈ N. L-fuzzy ternary subnear-rings 2049

μ is called an L-fuzzy ideal of N if μ is an L-fuzzy ternary subnear-ring of N and

(iii) μ(y + x − y) ≥ μ(x),

(iv) μ(xyz) ≥ μ(z) and

(v) μ((x + n)yz − xyz) ≥ μ(n), for all n, x, y, z ∈ N.

Note that μ is an L-fuzzy left ideal of N if it satisfies (i), (ii), (iii) and (iv), and μ is an L-fuzzy right ideal of N if it satisfies (i), (ii), (iii) and (v).

Example 4.2 Let N = {a, b, c, d}. Consider the ternary near-ring (N,+, ·) defined in Example 3.4. Define μ : N → L by μ(c)=μ(d) <μ(b) <μ(a). Then μ is an L-fuzzy ideal of N.

Lemma 4.3 ([12]) If an L-fuzzy subset μ of N satisfies the property (i) in Definition 4.1, then

(i) μ(0) ≥ μ(x) and

(ii) μ(−x)=μ(x), for all x ∈ N.

Theorem 4.4 Let N be a ternary near-ring and μ ∈ F (N). The following statements are satisfied.

(i) μ is an L-fuzzy ternary subnear-ring of N if and only if for any t ∈ L such that μt = ∅, μt is a ternary subnear-ring of N.

(ii) μ is an L-fuzzy left (resp. right) ideal of N if and only if for any t ∈ L such that μt = ∅, μt is a left (resp. right) ideal of N.

(iii) μ is an L-fuzzy ideal of N if and only if for any t ∈ L such that μt = ∅, μt is an ideal of N.

Proof. (i) Let μ be an L-fuzzy ternary subnear-ring of N. Let t ∈ L such that μt = ∅. Let x, y, z ∈ μt. Then μ(x),μ(y),μ(z) ≥ t. Then μ(x − y) ≥ inf{μ(x),μ(y)}≥t and μ(xyz) ≥ inf{μ(x),μ(y),μ(z)}≥t.Sox−y, xyz ∈ μt. Hence μt is a ternary subnear-ring of N. Conversely, let x, y, z ∈ N and t = inf{μ(x),μ(y)}. Then μ(x),μ(y) ≥ t.Thusx, y ∈ μt. By assumption, x − y ∈ μt.Soμ(x − y) ≥ t = inf{μ(x),μ(y)}. Next, let s = inf{μ(x),μ(y),μ(z)}. Then μ(x),μ(y),μ(z) ≥ s.Thusx, y, z ∈ μs. By assumption, xyz ∈ μs. So μ(xyz) ≥ s = inf{μ(x),μ(y),μ(z)}. Therefore, μ is an L-fuzzy ternary subnear-ring of N. (ii) Let μ be an L-fuzzy left ideal of N. Let t ∈ L such that μt = ∅. Let n ∈ N, and let x ∈ μt. Then μ(x) ≥ t.Soμ(n + x − n) ≥ μ(x) ≥ t. 2050 W. Nakkhasen and B. Pibaljommee

Then n + x − n ∈ μt. Then (μt, +) is a normal subgroup of (N,+). Let y ∈ NNμt. Then y = n1n2z for some n1,n2 ∈ N and for some z ∈ μt. Then μ(y)=μ(n1n2z) ≥ μ(z) ≥ t.Soy ∈ μt. Then NNμt ⊆ μt. Hence μt is a left ideal of N. Suppose that μ be an L-fuzzy right ideal of N. Let i ∈ μt, and let a, b, c ∈ N. Then μ((a + i)bc − abc) ≥ μ(i) ≥ t.So(a + i)bc − abc ∈ μt. Thus μt is a right ideal of N. Conversely, let x, y, z ∈ N. Let t ∈ L such that t = μ(x). Then x ∈ μt. Since (μt, +) is a normal subgroup of (N,+), y + x − y ∈ μt.Soμ(y + x − y) ≥ t = μ(x). Let s ∈ L such that s = μ(z). Then z ∈ μs. Since μs is a left ideal of N, xyz ∈ μs.Soμ(xyz) ≥ s = μ(z). Thus μ is an L-fuzzy left ideal of N. Suppose that μt is a right ideal of N.   Let n ∈ N, and let t ∈ L such that μ(n)=t . Then (x + n)yz − xyz ∈ μt ,so μ((x + n)yz − xyz) ≥ t = μ(n). Thus μ is an L-fuzzy right ideal of N. (iii) It follows by (ii).

Theorem 4.5 Let N be a ternary near-ring. The following statements are satisfied. (i) If A is a ternary subnear-ring of N, then there exists an L-fuzzy ternary subnear-ring μ of N such that μt = A for some t ∈ L. (ii) If A is a left (resp. right) ideal of N, then there exists an L-fuzzy left (resp. right) ideal μ of N such that μt = A for some t ∈ L. (iii) If A is an ideal of N, then there exists an L-fuzzy ideal μ of N such that μt = A for some t ∈ L.

Proof. Let t ∈ L and define an L-fuzzy set of N by t if x ∈ A, μ(x)= 0 otherwise.

It follows that μt = A. (i) Assume that A is a ternary subnear-ring of N.It is clear that every nonempty level subset of μ is a ternary subnear-ring of N. By Theorem 4.4 (i), we have μ is an L-fuzzy ternary subnear-ring of N. (ii) Let A be a left ideal of N and let x, y, z ∈ N. Suppose that μ(y + x − y) < μ(x). Since | Im(μ) |=2,μ(y + x − y) = 0 and μ(x)=t.Soy + x − y ∈ A and x ∈ A. But (A, +) is a normal subgroup of (N,+), then y + x − y ∈ A. This is a contradiction. Hence μ(y + x − y) ≥ μ(x). Similarly, μ(xyz) ≥ μ(z). Thus μ is an L-fuzzy left ideal of N. Suppose that A is a right ideal of N and assume that μ((x + i)yz − xyz) <μ(i) for some x, y, z ∈ N and i ∈ A. Since | Im(μ) |=2,μ((x+i)yz−xyz) = 0 and μ(i)=t and hence (x+i)yz−xyz ∈ A and i ∈ A. Since A is a right ideal of N,(x + i)yz − xyz ∈ A, which leads to a contradiction. Thus μ is an L-fuzzy right ideal of N. (iii) It is true by (ii). L-fuzzy ternary subnear-rings 2051

Let N be a ternary near-ring. If μ is an L-fuzzy ternary subnear-ring of N,we call μt(= ∅)alevel ternary subnear-ring of μ. The level left ideals (resp. right ideals, ideals) of μ are defined analogously. The following theorem is easy to verify.

Theorem 4.6 Let N be a ternary near-ring and μ ∈ F (N).Ifμ is an L-fuzzy subset of N, then two level subsets μs and μt (with s

Let N be a ternary near-ring. For any μ ∈ F (N), we denote by Im(μ) the image set of μ. The following theorem is easy to verify.

Theorem 4.7 Let N be a ternary near-ring. The following statements are satisfied.

(i) Let μ be an L-fuzzy subset of N.IfIm(μ)={t1,t2,... ,tn}, where { | ≤ ≤ } t1 < ... < tn, then μti 1 i n is the collection of all level subsets of μ.

(ii) Let μ be an L-fuzzy ternary subnear-ring of N.IfIm(μ)={t1,t2,... ,tn}, { | ≤ ≤ } where t1 < ... < tn, then μti 1 i n is the collection of all level ternary subnear-rings of μ.

(iii) Let μ be an L-fuzzy left (resp. right) ideal of N.IfIm(μ)={t1,t2,... ,tn}, { | ≤ ≤ } where t1 < ... < tn, then μti 1 i n is the collection of all level left (resp. right) ideals of μ.

(iv) Let μ be an L-fuzzy ideal of N.IfIm(μ)={t1,t2,... ,tn}, where t1 < { | ≤ ≤ } ...

Theorem 4.8 Let N and R be ternary near-rings and ϕ : N → R be a homomorphism. The following statements are satisfied.

(i) Let μ be an L-fuzzy ternary subnear-ring of ϕ(N). Then the preimage of μ under ϕ is an L-fuzzy ternary subnear-ring of N.

(ii) Let μ be an L-fuzzy left (resp. right) ideal of ϕ(N). Then the preimage of μ under ϕ is an L-fuzzy left (resp. right) ideal of N.

(iii) Let μ be an L-fuzzy ideal of ϕ(N). Then the preimage of μ under ϕ is an L-fuzzy ideal of N. 2052 W. Nakkhasen and B. Pibaljommee

Proof. (i) Let μ be an L-fuzzy ternary subnear-ring of F (ϕ(N)) and ν be the preimage of μ under ϕ. Then for any x, y, z ∈ N, ν(x − y)=μ(ϕ(x − y)) = μ(ϕ(x) − ϕ(y)) ≥ inf{μ(ϕ(x)),μ(ϕ(y))} = inf{ν(x),ν(y)} and ν(xyz)=μ(ϕ(xyz)) = μ(ϕ(x)ϕ(y)ϕ(z)) ≥ inf{μ(ϕ(x)),μ(ϕ(y)),μ(ϕ(z))} = inf{ν(x),ν(y),ν(z)}. Then ν is an L-fuzzy ternary subnear-ring of N. (ii) Let μ be an L-fuzzy left ideal of F (ϕ(N)) and ν be the preimage of μ under ϕ. Then for any n, x, y, z ∈ N, we have ν(y + x − y)=μ(ϕ(y + x − y)) = μ(ϕ(y)+ϕ(x) − ϕ(y)) ≥ μ(ϕ(x)) = ν(x) and ν(xyz)=μ(ϕ(xyz)) = μ(ϕ(x)ϕ(y)ϕ(z)) ≥ μ(ϕ(z)) = ν(z). This shows that ν is an L-fuzzy left ideal of N. Suppose that μ is an L-fuzzy right ideal of F (ϕ(N)) and ν is the preimage of μ under ϕ. Then ν((x + n)yz − xyz)=μ(ϕ((x + n)yz − xyz)) = μ((ϕ(x)+ϕ(n))ϕ(y)ϕ(z) − ϕ(x)ϕ(y)ϕ(z)) ≥ μ(ϕ(n)) = ν(n). Then ν is an L-fuzzy right ideal of N. (iii) follows from (ii).

We say that an L-fuzzy set μ of N has the sup property if, for any subset T of N, there exists t0 ∈ T such that μ(t0) = sup μ(t). t∈T Theorem 4.9 Let ϕ : N → R be a ternary near-ring homomorphism, μ ∈ F (N) and ν be the image of μ under ϕ.Ifμ is an L-fuzzy left (resp. right) ideal of N having the sup property, then ν is an L-fuzzy left (resp. right) ideal of ϕ(N). Proof. Let ϕ : N → R be a ternary near-ring homomorphism and μ be an L-fuzzy left ideal of N with the sup property and ν be the image of μ under −1 −1 ϕ. Given ϕ(x), ϕ(y), ϕ(z) ∈ ϕ(N), let x0 ∈ ϕ (ϕ(x)), yo ∈ ϕ (ϕ(y)) and −1 z0 ∈ ϕ (ϕ(z)) such that

μ(x0) = sup μ(t),μ(y0) = sup μ(t) and μ(z0) = sup μ(t), t∈ϕ−1(ϕ(x)) t∈ϕ−1(ϕ(y)) t∈ϕ−1(ϕ(z)) respectively. Then

ν(ϕ(x) − ϕ(y)) = sup μ(t) ≥ μ(x0 − y0) ≥ inf{μ(x0),μ(y0)} t∈ϕ−1(ϕ(x)−ϕ(y)) = inf{ sup μ(t), sup μ(t)} t∈ϕ−1(ϕ(x)) t∈ϕ−1(ϕ(y)) = inf{ν(ϕ(x)),ν(ϕ(y))}, L-fuzzy ternary subnear-rings 2053

ν(ϕ(x)ϕ(y)ϕ(z)) = sup μ(t) ≥ μ(x0y0z0) ≥ inf{μ(x0)μ(y0)μ(z0)} t∈ϕ−1(ϕ(x)ϕ(y)ϕ(z)) = inf{ sup μ(t), sup μ(t), sup μ(t)} t∈ϕ−1(ϕ(x)) t∈ϕ−1(ϕ(y)) t∈ϕ−1(ϕ(z)) = inf{ν(ϕ(x)),ν(ϕ(y)),ν(ϕ(z))},

ν(ϕ(y)+ϕ(x) − ϕ(y)) = sup μ(t) ≥ μ(y0 + x0 − y0) ≥ μ(x0) t∈ϕ−1(ϕ(y)+ϕ(x)−ϕ(y)) = sup μ(t)=ν(ϕ(x)) and t∈ϕ−1(ϕ(x))

ν(ϕ(x)ϕ(y)ϕ(z)) = sup μ(t) ≥ μ(x0y0z0) ≥ μ(z0) t∈ϕ−1(ϕ(x)ϕ(y)ϕ(z)) = sup μ(t)=ν(ϕ(z)). t∈ϕ−1(ϕ(z))

Then ν is an L-fuzzy left ideal of ϕ(N). Assume that μ is an L-fuzzy right ideal of N. Let ϕ(i) ∈ ϕ(N), and let −1 i0 ∈ ϕ (ϕ(i)) such that μ(i0) = sup ϕ(t). Then t∈ϕ−1(ϕ(i))

ν((ϕ(x)+ϕ(i))ϕ(y)ϕ(z) − ϕ(x)ϕ(y)ϕ(z))

= sup μ(t) ≥ μ((x0 + i0)y0z0 − x0y0z0) t∈ϕ−1((ϕ(x)+ϕ(i))ϕ(y)ϕ(z)−ϕ(x)ϕ(y)ϕ(z))

≥ μ(i0) = sup ϕ(t)=ν(ϕ(i)). t∈ϕ−1(ϕ(i))

Then ν is an L-fuzzy right ideal of ϕ(N).

Theorem 4.10 Let N and R be ternary near-rings and ϕ : N → R be a homomorphism. The following statements are satisfied.

(i) Let μ be an L-fuzzy ternary subnear-ring of N. Then the homomorphic image ϕ(μ) of μ under ϕ is an L-fuzzy ternary subnear-ring of R.

(ii) Let μ be an L-fuzzy left (resp. right) ideal of N. Then the homomorphic image ϕ(μ) of μ under ϕ is an L-fuzzy left (resp. right) ideal of R.

(iii) Let μ be an L-fuzzy ideal of N. Then the homomorphic image ϕ(μ) of μ under ϕ is an L-fuzzy ideal of R.

Proof. (i) By Theorem 4.4 (i), it is sufficient to show that each nonempty level subset of ϕ(μ) is a ternary subnear-ring of R. Let t ∈ L such that 2054 W. Nakkhasen and B. Pibaljommee

(ϕ(μ))t = ∅.Ift = 0, then (ϕ(μ))t = R. Assume that t = 0. By Proposition 2.4, (ϕ(μ))t = ϕ(μt−s). 0

Then ϕ(μt−s) = ∅ for all 0

Definition 4.11 A ternary subnear-ring A of a ternary near-ring N is said to be characteristic if ϕ(A)=A for all ϕ ∈ Aut(N) where Aut(N) is the set of all automorphisms of N. The characteristic left ideals (resp. right ideals, ideals) are defined analogously.

Definition 4.12 An L-fuzzy ternary subnear-ring μ of N is said to be L- fuzzy characteristic ternary subnear-ring if μ(ϕ(x)) = μ(x) for all x ∈ N and ϕ ∈ Aut(N). The L-fuzzy characteristic left ideals (resp. right ideals, ideals) are defined analogously.

Theorem 4.13 Let N and R be ternary near-rings, ϕ : N → R a homo- morphism and μ ∈ F (R). Define μϕ ∈ F (N) by μϕ(x)=μ(ϕ(x)) for all x ∈ N. The following statements are satisfied.

(i) If μ is an L-fuzzy ternary subnear-ring of R, then μϕ is an L-fuzzy ternary subnear-ring of N.

(ii) If μ is an L-fuzzy left (resp. right) ideal of R, then μϕ is an L-fuzzy left (resp. right) ideal of N.

(iii) If μ is an L-fuzzy ideal of R, then μϕ is an L-fuzzy ideal of N.

Proof. (i) Let x, y, z ∈ N. We have

μϕ(x − y)=μ(ϕ(x − y)) = μ(ϕ(x) − ϕ(y)) ≥ inf{μ(ϕ(x)),μ(ϕ(y))} = inf{μϕ(x),μϕ(y)} and μϕ(xyz)=μ(ϕ(xyz)) = μ(ϕ(x)ϕ(y)ϕ(z)) ≥ inf{μ(ϕ(x)),μ(ϕ(y)),μ(ϕ(z))} = inf{μϕ(x),μϕ(y),μϕ(z)}.

Therefore, μϕ is an L-fuzzy ternary subnear-ring of N. The proofs of (ii) and (iii) are similar to the proof of (i). L-fuzzy ternary subnear-rings 2055

Theorem 4.14 Let N be a ternary near-ring. The following statements are satisfied.

(i) If μ is an L-fuzzy characteristic ternary subnear-ring of N, then each level ternary subnear-ring of μ is characteristic.

(ii) If μ is an L-fuzzy characteristic left (resp. right) ideal of N, then each level left (resp. right) ideal of μ is characteristic.

(iii) If μ is an L-fuzzy characteristic ideal of N, then each level ideal of μ is characteristic.

Proof. (i) Let μ be an L-fuzzy characteristic ternary subnear-ring of N, ϕ ∈ Aut(N) and t ∈ L.Ify ∈ ϕ(μt). Then there exists x ∈ μt such that ϕ(x)=y and so μ(y)=μ(ϕ(x)) = μ(x) ≥ t.Thusy ∈ μt. Conversely, if y ∈ μt and since ϕ ∈ Aut(N), then y = ϕ(x) for some x ∈ N and μ(x)= μ(ϕ(x)) = μ(y) ≥ t. It follows that y ∈ ϕ(μt). Thus ϕ(μt)=μt. Then μt is characteristic. The proofs of (ii) and (iii) are similar to the proof of (i).

The following theorem is the converse of Theorem 4.14.

Theorem 4.15 Let N be a ternary near-ring and μ ∈ F (N). The following statements are satisfied.

(i) If each level ternary subnear-ring of μ is characteristic, then μ is an L-fuzzy characteristic ternary subnear-ring of N.

(ii) If each level left (resp. right) ideal is characteristic, then μ is an L-fuzzy characteristic left (resp. right) ideal of N.

(iii) If each level ideal is characteristic, then μ is an L-fuzzy characteristic ideal of N.

Proof. (i) Let x ∈ N, ϕ ∈ Aut(N) and t = μ(x). Then x ∈ μt and x ∈ μs for all s ∈ L with s>t. Since each level ternary subnear-ring of μ is characteristic, ϕ(x) ∈ ϕ(μt)=μt.Thusμ(ϕ(x)) ≥ t. Suppose μ(ϕ(x)) = r> t. Then ϕ(x) ∈ μr = ϕ(μr). This implies that x ∈ μr, a contradiction. Hence μ(ϕ(x)) = μ(x). Therefore μ is an L-fuzzy characteristic ternary subnear-ring of N. The proofs of (ii) and (iii) are similar to the proof of (i). 2056 W. Nakkhasen and B. Pibaljommee

5 Normal L-fuzzy ternary subnear-rings

In this section, we apply the concept of normal L-fuzzy ideals of semirings introduced by Jun, Neggers and Kim ([7]) to introduce the normal L-fuzzy subset of a ternary near-ring where L is a complemented distributive lattice and study some of their fundamental properties. First of all we review the notion of complemented lattice defined in [5]. Let L be a bounded lattice with the greatest element 1 and the least element 0 and a, b ∈ L. An element a of L is called a complement of b if a ∧ b = 0 and a ∨ b =1.Acomplemented lattice is a bounded lattice in which every element has a complement.

Lemma 5.1 ([5]) In a bounded distributive lattice L, an element of L can have only one complement.

All of this section we assume that L is a complemented distributive lattice with the greatest element 1 and the least element 0.

Definition 5.2 An L-fuzzy subset μ of a ternary near-ring N is called normal if μ(0) = 1.

Let L be a complemented distributive with the greatest element 1 and the least element 0. As a notation in [7] we denote x + y by x ∨ y and x − y by x ∧ y, where x, y ∈ L and y is the complement of y. Let N be a ternary near-ring and μ ∈ F (N). Define an L-fuzzy subset μ+ of N by μ+(x)=μ(x)+(1− μ(0)) = μ(x) ∨ μ(0), for all x ∈ N.

Proposition 5.3 Let N be a ternary near-ring and μ ∈ F (N). The following statements are satisfied.

(i) μ+ is a normal L-fuzzy subset of N containing μ.

(ii) (μ+)+ = μ+.

(iii) μ is normal if and only if μ = μ+.

Proof. (i) We can see that μ+(0) = μ(0)∨μ(0) = 1. Then for every x ∈ N, μ(x) ≤ μ+(x). i.e., μ contains in μ+. (ii) By (i), we have (μ+)+(x)=μ+(x) ∨ μ+(0) = μ+(x) ∨ 0=μ+(x). This means that (μ+)+ = μ+. (iii) Assume that μ is normal. Then μ+(x)=μ(x) ∨ μ(0) = μ(x) ∨ 1 = μ(x) ∨ 0=μ(x). The converse is obvious by (i). L-fuzzy ternary subnear-rings 2057

Theorem 5.4 Let N be a ternary near-ring and μ ∈ F (N). The following statements are satisfied. (i) If μ is an L-fuzzy ternary subnear-ring of N, then μ+ is a normal L-fuzzy ternary subnear-ring of N containing μ. (ii) If μ is an L-fuzzy left (resp. right) ideal of N, then μ+ is a normal L-fuzzy left (resp. right) ideal of N containing μ. (iii) If μ is an L-fuzzy ideal of N, then μ+ is a normal L-fuzzy ideal of N containing μ.

Proof. (i) Let x, y, z ∈ N. Then

μ+(x − y)=μ(x − y) ∨ μ(0) ≥ (μ(x) ∧ μ(y)) ∨ μ(0) =(μ(x) ∨ μ(0)) ∧ (μ(y) ∨ μ(0))=μ+(x) ∧ μ+(y) and μ+(xyz)=μ(xyz) ∨ μ(0) ≥ (μ(x) ∧ μ(y) ∧ μ(z)) ∨ μ(0) =(μ(x) ∨ μ(0)) ∧ (μ(y) ∨ μ(0)) ∧ (μ(z) ∨ μ(0)) = μ+(x) ∧ μ+(y) ∧ μ+(z).

Hence μ+ is an L-fuzzy ternary subnear-ring of N. By Proposition 5.3 (i), we have that μ+ is a normal L-fuzzy ternary subnear-ring of N containing μ. The proofs of (ii) and (iii) are similar to the proof of (i).

Let Sub(N) be the set of all ternary subnear-rings of N. Then (Sub(N), ∧, ∨) is a complete lattice (see [15]), where {Ai ∈ Sub(N) | i ∈ I} = Ai and i∈I {Ai ∈ Sub(N) | i ∈ I} = {A ∈ Sub(N) | A ⊇ Ai}. i∈I

We denote FS(N) by the set of all L-fuzzy ternary subnear-rings of N. Then (FS(N), ∧, ∨) forms the complete lattice, where {μi ∈ FS(N) | i ∈ I} = inf{μi ∈ FS(N) | i ∈ I} and {μi ∈ FS(N) | i ∈ I} = {ν ∈ FS(N) | ν ⊇ μi}. i∈I

It turn out that Sub(N) can be embedded into FS(N) ([15]). We denote FSN(N) by the set of all normal L-fuzzy ternary subnear-rings of N, i.e., FSN(N)={μ ∈ FS(N) | μ(0) = 1}.

Theorem 5.5 FSN(N) is a complete sublattice of FS(N). 2058 W. Nakkhasen and B. Pibaljommee

Proof. It is clear that FSN(N) ⊆ FS(N). Let A ⊆ FSN(N) such that A = ∅. We have that μ ∈ FS(N) and μ ∈ FS(N). Then μ(0) = μ∈A μ∈A μ∈A inf{μ(0) | μ ∈ A} =1.So μ ∈ FSN(N). Let ν0 = μ. Then ν0(0) = μ∈A μ∈A μ(0) = {ν(0) ∈ FS(N) | ν(0) ⊇ μ(0)}. Since 1 = μ(0) ≤ ν0(0), μ∈A μ∈A μ∈A ν0(0) = 1. Hence μ ∈ FSN(N). Therefore, FSN(N) is a complete μ∈A sublattice of FS(N).

References

[1] S. Abou-Zaid, On fuzzy subnear-ring and ideals, Fuzzy Sets and Systems, 44 (1991), 139-146.

[2] R. Chinram and S. Malee, L-fuzzy ternary subsemirings and L-fuzzy ideals in ternary semirings, IAENG International Journal of Applied Mathemat- ics, 40(3) (2010), 124-129.

[3] V. N. Dixit, R. Kumar and N. Ajal, On fuzzy ring, Fuzzy Sets and Systems, 49 (1992), 205-213.

[4] T. K. Dutta and S. Kar, On regular ternary semirings, Advances in , proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific, (2003), 343-355.

[5] G. Gr¨atzer, General lattice theory, Academic Press, Inc., New York, 1978.

[6] S. M. Hong, Y. B. Jun and H. S. Kim, Fuzzy ideals in near-ring, Bull. Korean Math. Soc., 35(3) (1998), 455-464.

[7] Y. B. Jun, J. Neggers and H. S. Kim, Normal L-fuzzy ideals in semirings, Fuzzy Sets and Systems, 82 (1996), 383-386.

[8] Y. B. Jun, J. Neggers and H. S. Kim, On L-fuzzy ideals in semirings I, Czechoslovak Mathematical Journal 48(123) (1998), 669-675.

[9] Y. B. Jun, M. Sapacai and M. A. Ozt¨¨ urk, Fuzzy ideals in gamma near-ring, Tr. J. of Mathematics, 22 (1998), 449-459.

[10] J. Kavikumae and A. B. Khamis, Fuzzy ideals and fuzzy quasi-ideals in ternary semirings, IAENG International Journal of Applied Mathematics, 37(2) (2007), 102-106. L-fuzzy ternary subnear-rings 2059

[11] Kavikumae, A. Khamis and Y. B. Jun, Fuzzy bi-ideals in ternary semir- ings, International Journal of Computational and Mathematical, 3(4) (2009), 160-164.

[12] S. D. Kim and H. S. Kim, On fuzzy ideals of near-rings, Bull. Korean Math. Soc., 33(4) (1996), 593-601.

[13] R. Kumar, Fuzzy irreducible ideals in rings, Fuzzy Sets and Systems, 42 (1991), 369-379.

[14] R. Kumar, Certain fuzzy ideals of rings redefined, Fuzzy Sets and Systems, 46 (1992), 251-260.

[15] T. Kuraoka and N. Y. Suzuki, Lattice of fuzzy subalgebras in universal algebra, Algebra Universalis, 47 (2002), 223-237.

[16] D. H. Lehmer, A ternary analogue of abelian groups, American Journal of Mathematics, 59 (1932), 329-339.

[17] W. G. Lister, Ternary rings, Transaction of American Mathematical So- ciety, 154 (1971), 37-55.

[18] W. Liu, Fuzzy invariant subgroups and fuzzy ideal, Fuzzy Sets and Sys- tems, 8 (1982), 133-139.

[19] S. Malee and R. Chinram, k-fuzzy ideals of ternary semirings, World Academy of Science, Engineering and Technology, 67 (2010), 485-489.

[20] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.

[21] F. M. Sioson, Ideal theory in ternary , Math. Japon., 10 (1965), 63-84.

[22] Z. Yue, L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Sets and Systems, 27 (1988), 345-350.

[23] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.

Received: March, 2012