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

Using the ideas of belonging and quasi-coincidence of a fuzzy point with a fuzzy set, the concepts of (α,β)(α,β)-fuzzy ideals and (α,β)(α,β)-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 (∈,∈∨q)(∈,∈∨q)-fuzzy left ideals, (∈,∈∨q)(∈,∈∨q)-fuzzy quasi-ideals and (∈,∈∨q)(∈,∈∨q)-fuzzy generalized bi-ideals are given.