International Mathematical Forum, Vol. 7, 2012, no. 41, 2045 - 2059 L-Fuzzy Ternary Subnear-Rings1 Warud Nakkhasen2 and Bundit Pibaljommee3 Department of Mathematics, 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-ring, 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 lattice 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 distributive lattice. 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 group 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 semiring which is a generalization of the ternary ring introduced by Lister ([17]). The notion of fuzzy ideals and fuzzy quasi-ideals in ternary semirings 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 semigroup 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 algebraic structure (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<s<t Given any two sets X and Y , let μ ∈ F (X) and let f : X → Y be a function. Define ν ∈ F (Y ) by for y ∈ Y , ⎧ ⎨ sup μ(x)iff −1(y) = ∅, x∈f −1(y) ν(y)=⎩ 0 otherwise. 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.