Left and right regular elements of the semigroups of transformations preserving an equivalence relation and a cross-section
Nares Sawatraksa, Chaiwat Namnak, Ronnason Chinram · Asian-European Journal of Mathematics · 2018
Let [Formula: see text] be the semigroup of all transformations on a set [Formula: see text]. For an arbitrary equivalence relation [Formula: see text] on [Formula: see text] and a cross-section [Formula: see text] of the partition [Formula: see text] induced by [Formula: see text], let [Formula: see text] [Formula: see text] Then [Formula: see text] and [Formula: see text] are subsemigroups of [Formula: see text]. In this paper, we characterize left regular, right regular and completely regular elements of [Formula: see text] and [Formula: see text]. We also investigate conditions for which of these semigroups to be left regular, right regular and completely regular semigroups.