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 $(\in ,\in \vee q)$ ‐fuzzy left ideals, $(\in ,\in \vee q)$ ‐fuzzy quasi‐ideals and $(\in ,\in \vee q)$ ‐fuzzy generalized bi‐ideals are given.