tailieunhanh - On intuitionistic fuzzy bi-ideals of semigroups
In this paper, we consider the intuitionistic fuzzification of the concept of several ideals in a semigroup S, and investigate some properties of such ideals. | Turk J Math 29 (2005) , 201 – 210. ¨ ITAK ˙ c TUB On Intuitionistic Fuzzy Bi-Ideals of Semigroups K. H. Kim, J. G. Lee Abstract We consider the intuitionistic fuzzification of the concept of several ideals in a semigroup S, and investigate some properties of such ideals. Key Words: Intuitionistic fuzzy (1, 2)-ideal, intuitionistic fuzzy bi-ideal, intuitionistic fuzzy ideal. 1. Introduction After the introduction of fuzzy sets by L. A. Zadeh [8], several researchers explored on the generalization of the the notion of fuzzy set. The concept of intuitionistic fuzzy set was introduced by K. T. Atanassov [1, 2], as a generalization of the notion of fuzzy set. In [3], N. Kuroki gave some properties of fuzzy ideals and fuzzy bi-ideals in semigroups. The concept of (1, 2)-ideals in semigroups was introduced by S. Lajos [5]. In this paper, we consider the intuitionistic fuzzification of the concept of several ideals in a semigroup S, and investigate some properties of such ideals. 2. Preliminaries Let S be a semigroup. By a subsemigroup of S we mean a non-empty subset A of S such that A2 ⊆ A, and by a left (right) ideal of S we mean a non-empty subset A of S such that SA ⊆ A (AS ⊆ A). By two-sided ideal or simply ideal, we mean a non-empty subset of S which is both a left and a right ideal of S. A subsemigroup A of a semigroup 2000 Mathematics Subject Classification: 20M12, 04A72. 201 KIM, LEE S is called a bi-ideal of S if ASA ⊆ A. A subsemigroup A of S is called a (1, 2)-ideal of S if ASA2 ⊆ A. A semigroup S is said to be (2, 2)-regular if x ∈ x2 Sx2 for any x ∈ S. A semigroup S is said to be regular if, for each x ∈ S, there exists y ∈ S such that x = xyx. A semigroup S is said to be completely regular if, for each x ∈ S, there exists y ∈ S such that x = xyx and xy = yx. For a semigroup S, note that S is completely regular if and only if S is a union of groups if and only if S is (2, 2)-regular. A semigroup S is said to be left (resp. right) duo if every .
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