ELSEVIER Fuzzy Sets and Systems 92 (1997) 103-111 FUZZY sets and systems Fuzzy ideals and fuzzy bi-ideals in fuzzy semigroups K.A. Dib*, N. Galhum Department o[ Mathematics, Faculty of Science, Fayoum Branch. Cairo Unit~ersity. Fayoum. Et, typt Received January 1994; revised June 1996 Abstract The purpose of this paper is to introduce some basic concepts of fuzzy algebra such as fuzzy (left, right) ideal and fuzzy bi-ideal in fuzzy semigroup, through the new approach of fuzzy space and fuzzy group introduced by Dib (1994). Our notion of fuzzy ideal and fuzzy bi-ideal includes the (classical) concepts of fuzzy ideal and fuzzy bi-ideal of ordinary semigroup. Many counterexamples are also given. © 1997 Published by Elsevier Science B.V. Keywords: Fuzzy space; Fuzzy semigroup; Fuzzy ideal; Fuzzy bi-ideal O. Introduction The study of the fuzzy algebraic structures has started with the introduction of the concepts of fuzzy (subgroupoids) subgroups and fuzzy (left, fight) ideals in the pioneering paper of Rosenfeld [9]. In 1975, Negoita and Ralescu [8] considered a gen- eralization of Rosenfeld's definition where the unit interval [0, 1] is replaced by an appropriate lattice structure. In 1979, Anthony and Sherwood [1] re- defined fuzzy (subgroupoids) subgroups using the concept of triangular norm. Kuroki [4, 5] introduced and studied fuzzy (left, right) ideals and fuzzy bi- ideals in semigroups. Several mathematicians [6-8, 10] have followed the Rosenfeld approach in in- vestigating fuzzy algebra where a given ordinary algebraic structure on a given set X is assumed, then introducing the fuzzy algebraic structure as a fuzzy subset A of X satisfying some suitable conditions. * Correspondingauthor. 0165-0114/97/$17.00 (~) 1997 Published by ElsevierScience B.V. All PHS0165-0114(96)00170-4 In this paper, any fuzzy algebraic structure in the Rosenfeld approach is called a classical fuzzy struc- ture. In 1991, new concepts of fuzzy Cartesian prod- uct, fuzzy relation, fuzzy function and a fuzzy binary operation on a set have been introduced [3]. As we know, the concept of universal set played an essential role in ordinary mathematics, e.g. any algebraic sys- tem is a universal set with one or more binary oper- ations and the metric space is a universal set, having distance, etc. Therefore, in the absence of the concept of fuzzy universal set, the formulation of a concrete definition of any fuzzy structure is not evident. Re- cently, the concept of a fuzzy space is introduced in [2] as a replacement for the concept of universal set in the ordinary case. Moreover, a fuzzy subspace, fuzzy bi- nary operation on a fuzzy space and fuzzy (subgroup) group were introduced also in [2]. In the present paper, we introduce some basic con- cepts of fuzzy algebra such as fuzzy (left, fight) ideals and fuzzy bi-ideals in fuzzy semigroup, through the new approach of fuzzy spaces and fuzzy groups intro- duced in [2]. A relation between the introduced fuzzy (left, right) ideals or fuzzy bi-ideals and the classical rights reserved