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
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