Fuzzy Bi-ideals in Ternary Semirings Kavikumar, Azme Khamis, and Young Bae Jun, Abstract—The purpose of the present paper is to study the concept of fuzzy bi-ideals in ternary semirings. We give some characteriza- tions of fuzzy bi-ideals. Characterizations of regular ternary semirings are provided. Keywords—Fuzzy ternary subsemiring, fuzzy quasi-ideal, fuzzy bi-ideal, regular ternary semiring I. I NTRODUCTION T ERNARY semirings are one of the generalized structures of semirings. The notion of ternary algebraic system was introduced by Lehmer [8]. He investigated certain ternary algebraic systems called triplexes which turn out to be commu- tative ternary groups. Dutta and Kar [1] introduced the notion of ternary semiring which is a generalization of the ternary ring introduced by Lister [9]. Good and Hughes [3] introduced the notion of bi-ideal and Steinfeld [11], [12] introduced the notion of quasi-ideal. In 2005, Kar [5] studied quasi-ideals and bi-ideals of ternary semirings. Ternary semiring arises naturally, for instance, the ring of integers Z is a ternary semiring. The subset Z + of all positive integers of Z forms an additive semigroup and which is closed under the ring product. Now, if we consider the subset Z − of all negative integers of Z, then we see that Z − is closed under the binary ring product; however, Z − is not closed under the binary ring product, i.e., Z − forms a ternary semiring. Thus, we see that in the ring of integers Z, Z + forms a semiring whereas Z − forms a ternary semiring. More generally; in an ordered ring, we can see that its positive cone forms a semiring whereas its negative cone forms a ternary semiring. Thus a ternary semiring may be considered as a counterpart of semiring in an ordered ring. The theory of fuzzy sets was first inspired by Zadeh [14]. Fuzzy set theory has been developed in many directions by many scholars and has evoked great interest among mathemati- cians working in different fields of mathematics. Rosenfeld [13] introduced fuzzy sets in the realm of group theory. Fuzzy ideals in rings were introduced by Liu [10] and it has been studied by several authors. Jun [4] and Kim and Park [7] have also studied fuzzy ideals in semirings. In 2007, [6] we have introduced the notions of fuzzy ideals and fuzzy quasi-ideals in ternary semirings. Our main purpose in this paper is to introduce the notions of fuzzy bi-ideal in ternary semirings and study regular ternary semiring in terms of these two subsystems of fuzzy subsemir- ings. We give some characteriztions of fuzzy bi-ideals. Kavikumar and Azme are with the Centre for Science Studies, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Johor, Malaysia e-mail: kaviphd@gmail.com. Y. B. Jun is with the Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Korea. Manuscript received April , ; revised II. PRELIMINARIES In this section, we review some definitions and some results which will be used in later sections. Definition 2.1. A set R together with associative binary operations called addition and multiplication (denoted by + and . respectively) will be called a semiring provided: (i) Addition is a commutative operation. (ii) there exists 0∈R such that a+ 0=a and a0=0a=0 for each a∈R, (iii) multiplication distributes over addition both from the left and the right. i.e., a(b + c)= ab + ac and (a + b)c = ac + bc Definition 2.2. A nonempty set S together with a binary operation, called addition and a ternary multiplication, denoted by juxtaposition, is said to be a ternary semiring if (S, +) is an additive commutative semigroup satisfying the following conditions: (i) (abc)de = a(bcd)e = ab(cde) (ii) (a + b)cd = acd + bcd (iii) a(b + c)d = abd + acd (iv) ab(c + d)= abc + abd, for all a, b, c, d, e ∈ S. Definition 2.3. (i) Let S be a ternary semiring. An additive subsemigroup T of S is called a ternary subsemiring of S if t 1 t 2 t 3 ∈ T , for all t 1 ,t 2 ,t 3 ∈ T . (ii) Let S be a ternary semiring.If there exists an element 0∈ S such that 0+a = a and 0ab = a0b=ab0=0 for all a, b ∈ S, then ”0” is called the zero element or simply the zero of the ternary semiring S. In this case we say that S is a ternary semiring with zero. (iii) Let A, B, C be three subsets of ternary semiring S. Then by ABC , we mean the set of all finite sums of the form ∑ a i b j c k with a i ∈ A, b j ∈ B,c k ∈ C. (iv) An additive subsemigroup I of S is called a left (resp., right, and lateral) ideal of S if s 1 s 2 i (resp.is 1 s 2 ,s 1 is 2 )∈ I , for all s 1 ,s 2 ∈ S and i ∈ I . If I is both left and right ideal of S, then I is called a two-sided ideal of S. If I is a left, a right and a lateral ideal of S, then I is called an ideal of S. An ideal I of S is called a proper ideal if I = S. Definition 2.4. (i) An additive subsemigroup (Q, +) of a ternary semiring S is called a quasi-ideal of S if QSS ∩ (SQS + SSQSS) ∩ SSQ ⊆ Q. (ii) An additive subsemigroup (Q, +) of a ternary semiring S is called a bi-ideal of S if QSQSQ ⊆ Q. Now, we review the concept of fuzzy sets [10], [13], [14]). Let X be a non-empty set. A map μ : X → [0, 1] is called a fuzzy set in X, and the complement of a fuzzy set μ in X, World Academy of Science, Engineering and Technology Vol:31 2009-07-22 991 International Science Index 31, 2009 waset.org/publications/9382