Characterization of some types of ordered semigroups in terms of fuzzy sets

It is well known that the right (left) regular, regular, and intra-regular ordered semigroups play an essential role in studying the structure, especially the decomposition, of ordered semigroups. In the present paper we study some more general classes containing the right regular, left regular, regular and intra-regular ordered semigroups. As an application of our results we get characterizations of right (left) regular, regular and intra-regular ordered semigroups in terms of fuzzy sets. We prove the following: An ordered semigroup S is right (resp. left) regular if and only if for each fuzzy subset f of S we have f ⊆ f2 ° 1 (resp. f ⊆ 1 ° f2). It is regular if and only if for each fuzzy subset f of S we have f ⊆ f ° 1 ° f, and it is intra-regular if and only if f ⊆ 1 ° f2 ° 1 for each fuzzy subset f of S (where 1 is the greatest element of the set of fuzzy subsets of S). Keeping in mind the definitions of right (left) regular, regular, and intra-regular ordered semigroups one can see how similar is the theory of ordered semigroups with the theory of fuzzy ordered semigroups.