Left regular and right regular elements of some semigroups

Abstract

We call an element x of a semigroup S a left [right] regular element if x = yx² [x = x²y] for some y ϵ S, or equivalently, x L x² [x R x²]. The variant of a semigroup S induced by a ϵ S is the semigroup (S,*) where x * y = xay for all x, y ϵ S. In this research, we characterize the left regular and right regular elements of some semigroups of transformations of sets and linear transformations. Moreover, the left regular and right regular elements of any variant of these semigroups are determined.