An (α,β)-Hesitant Fuzzy Set Approach to Ideal Theory in Semigroups

The aim of this manuscript is to introduce the \((\alpha,\beta)\)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of \((\alpha,\beta)\)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study \((\alpha,\beta)\)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of \((\alpha,\beta)\)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of \((\alpha,\beta)\)-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of \((\alpha,\beta)\)-hesitant fuzzy left ideals and \((\alpha,\beta)\)-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of \((\alpha,\beta)\)-hesitant ideals are given.