Regular ordered semigroups in terms of fuzzy subsets

Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mappingf : S -> [0, 1] where [0, 1] is the usual interval of real numbers. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadch. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids-semigroups have been already given by the same authors in [N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128-132; N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231-236; N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13-28] where S has been endowed with the structure of an ordered semigroup and defined fuzzy analogous for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals and quasi-ideals is well known as well. The characterization of regular le-semigroups (that is lattice ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of fuzzy ordered semigroups. Some of our results are true for ordered groupoids in general. (c) 2006 Elsevier Inc. All rights reserved.