A study of generalized fuzzy ideals in ordered semigroups

In mathematics, an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders, and error-correcting codes. A theory of fuzzy generalized sets on ordered semigroups can be developed. Using the notion of belongingness (is an element of) and quasi-coincidence (q) of fuzzy points with a fuzzy set, we introduce the concept of an (alpha, beta)-fuzzy left (resp. right) ideal of an ordered semigroup S, where alpha, beta is an element of {is an element of, q, is an element of boolean OR q, is an element of boolean AND q} with alpha not equal is an element of boolean AND q. Since the concept of (is an element of, is an element of boolean OR q)-fuzzy left (resp. right) ideals is an important and useful generalization of ordinary fuzzy left (resp. right) ideal, we discuss some fundamental aspects of (is an element of, is an element of boolean OR q)-fuzzy left (resp. right) ideals and ((is an element of) over bar, (is an element of) over bar boolean OR (q) over bar)-fuzzy left (resp. right) ideals. A fuzzy subset mu of an ordered semigroup S is an (is an element of, is an element of boolean OR q)-fuzzy left (resp. right) ideal if and only if mu(t), the level cut of mu is a left (resp. right) ideal of S, for all 0 < t <= B 0.5 and mu is an (<(is an element of)over bar>, (is an element of) over bar boolean OR (q) over bar)-fuzzy left (resp. right) ideal if and only if mu(t) is a left (resp. right) ideal of S, for all 0.5 < t <= 1. This means that an (is an element of, is an element of boolean OR q)-fuzzy left (resp. right) ideal and <(is an element of)over bar> boolean OR (q) over bar)-fuzzy left (resp. right) ideal are generalizations of the existing concept of fuzzy left (resp. right) ideals. Finally, we characterize regular ordered semigroups in terms of (is an element of, is an element of boolean OR q)-fuzzy left (resp. right) ideals.