East Asian Mathematical Journal Vol. 25 (2009), No. 4, pp. 449–467 ON INTERVAL VALUED FUZZY QUASI-IDEALS OF SEMIGROUPS N. Thillaigovindan and V. Chinnadurai Abstract. In this paper we shall introduce the notion of an i-v fuzzy interior ideal, an i-v fuzzy quasi-ideal and an i-v fuzzy bi-ideal in a semi- group. We study some properties of i-v fuzzy subsets and using their properties we characterize regular semigroups. 1. Introduction In 1975, Zadeh ([11]) introduced a new notion of fuzzy subsets viz., interval valued fuzzy subsets (in short, i-v fuzzy subsets) where the values of the mem- bership functions are closed intervals of numbers instead of a number. In ([3]), Biswas defined interval valued fuzzy subgroups and investigated some elemen- tary properties. Subsequently, Jun and Kim ([7]) and Davvaz ([4]) applied a few concept of i-v fuzzy subsets in near-rings. In this paper we introduce the notion of an i-v fuzzy interior ideal, an i-v fuzzy quasi-ideal and an i-v fuzzy bi-ideal in a semigroup. We investigate some of their properties. We give ex- amples which are i-v fuzzy interior ideal and i-v fuzzy bi-ideal but not i-v fuzzy ideal and i-v fuzzy quasi-ideal respectively. We find the equivalent conditions on which these i-v fuzzy subsets coincide. Finally we characterize regular semi- groups through their i-v fuzzy subsets. We also find equivalent conditions on regular semigroups through i-v fuzzy subsets. 2. Basic definitions and preliminary results Let S be a semi group. Let A and B be subsets of S, the multiplication of A and B is defined as AB = { ab ∈ S | a ∈ A and b ∈ B }. A nonempty subset A of S is called a subsemigroup of S if AA ⊆ A. A nonempty subset A of S is called a left (right)ideal of S if SA ⊆ A (AS ⊆ A). A is called a two-sided ideal (simply ideal ) of S if it is both a left and a right ideal of S. A nonempty subset A of S is called an interior ideal of S if SAS ⊆ A, and a quasi-ideal of S if AS ∩ SA ⊆ A. A subsemigroup A of S is called a bi-ideal of S if ASA ⊆ A. Received March 9, 2009; Accepted October 23, 2009. 2000 Mathematics Subject Classification. 18B40, 03E72, 08A72. Key words and phrases. regular, i-v fuzzy subsemigroup, i-v fuzzy ideal, i-v fuzzy interior ideal, i-v fuzzy fuzzy quasi-ideal, i-v fuzzy bi-ideal. c 2009 The Youngnam Mathematical Society 449