Various regularities for semigroups of transformations on a finite set determined by a zig-zag order

Abstract

Consider a finite ordered set [Formula: see text] with a zig-zag order. In this paper, we consider the semigroup [Formula: see text] of all compressing-preserving transformations on [Formula: see text]. Necessary and sufficient conditions for [Formula: see text] to be regular and coregular are given. Regular and coregular elements in [Formula: see text] are completely described. For a subsemigroup [Formula: see text] of the transformation semigroup [Formula: see text], we obtain that left regular and right regular elements in [Formula: see text] are identical. A characterization of left [right] regular elements in [Formula: see text] is given. We classify all left [right] regular semigroups [Formula: see text].