New types of fuzzy bi-ideals in ordered semigroups

In Jun et al. (Bull Malays Math Sci Soc (2) 32(3):391-408, 2009), (alpha, beta)-fuzzy bi-ideals are introduced and some characterizations are given. In this paper, we generalize the concept of (alpha, beta)-fuzzy bi-ideals and define (is an element of, is an element of boolean OR q(k))-fuzzy bi-ideals in ordered semigroups, which is a generalization of the concept of an (alpha, beta)-fuzzy bi-ideal in an ordered semigroup. Using this concept, some characterization theorems of regular, left (resp. right) regular and completely regular ordered semigroups are provided. In the last section, we give the concept of upper/lower parts of an (is an element of, is an element of boolean OR q(k))-fuzzy bi-ideal and investigate some interesting results of regular and intra-regular ordered semigroups.