On α-bases of semigroups

We introduce the notion of an [Formula: see text]-base of a semigroup, which generalizes the notions of bases of a semigroup, namely, a one-sided base, a bi-base, an interior base, a left weak-interior base, and a right weak-interior base. We characterize when a nonempty subset of a semigroup is an [Formula: see text]-base using the quasi-order defined by the principal [Formula: see text]-ideals of the semigroup. We also provide a characterization of conditions under which an [Formula: see text]-base of a semigroup is a subsemigroup of the semigroup. Moreover, we show that the cardinalities of any two [Formula: see text]-bases of an [Formula: see text]-regular semigroup are equal, where [Formula: see text] is the word in which 0 appears in the first or the last position.