Regular subsemigroups of the semigroups of transformations preserving a fence

Abstract

It is well-known that the transformation semigroup [Formula: see text] is regular, but their subsemigroups need not be. A fence is an ordered set that the order forms a path with alternating orientation. Consider [Formula: see text] as the base set of a fence [Formula: see text]. Two subsemigroups of [Formula: see text] are studied. Namely, the semigroup [Formula: see text] of all order-decreasing self-mappings of [Formula: see text] and the semigroup [Formula: see text] of all order-preserving self-mappings of [Formula: see text]. In this paper, we obtain that [Formula: see text] is a coregular subsemigroup of [Formula: see text]. A characterization of regular subsemigroups [Formula: see text] of [Formula: see text] is given, that is, [Formula: see text] is regular if and only if [Formula: see text]. Finally, we discuss the regularity of elements in [Formula: see text].