Characterizations of regular semigroups by (α, β)-fuzzy ideals

Using the ideas of belonging and quasi-coincidence of a fuzzy point with a fuzzy set, the concepts of (alpha, beta)-fuzzy ideals and (alpha, beta)-fuzzy generalized bi-ideals, which are generalization of fuzzy ideals and fuzzy generalized bi-ideals, in a semigroup are introduced, and related properties are investigated. We also define the lower and upper parts of fuzzy subsets of a semigroup. Characterizations of regular semigroups by the properties of the lower part of (is an element of, is an element of boolean OR q)-fuzzy left ideals, (is an element of, is an element of boolean OR q)-fuzzy quasi-ideals and (is an element of, is an element of boolean OR q)-fuzzy generalized bi-ideals are given. (C) 2009 Elsevier Ltd. All rights reserved.