Fuzzy bi-ideals in ordered semigroups

Given a set S, a fuzzy subset of S is, by definition, an arbitrary mapping f:S→[0,1] where [0,1] is the unit segment of the real line. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfelt, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. In the present paper we endow S with the structure of an ordered semigroup and define “fuzzy” analogous for several notions that have been proved to be useful in the theory of ordered semigroups. We define the fuzzy bi-ideals in ordered semigroups and we give the main theorem which characterizes the bi-ideals in terms of fuzzy bi-ideals. Then we characterize the left and right simple, the completely regular, and the strongly regular ordered semigroups by means of fuzzy bi-ideals. We also study the decomposition of left and right simple ordered semigroups and of ordered semigroups having the property a⩽a2 for all a, in terms of fuzzy bi-ideals. This decomposition is uniquely defined.