Order-preserving transformations with restricted range: regularity, Green’s relations, and ideals

Abstract

Let X be a chain and OT(X) the full order-preserving transformation semigroup on X. Let Y be a fixed nonempty subset of X and let OT(X, Y) be the subsemigroup of OT(X) of all order-preserving transformations with ranges contained in Y. In this paper, we investigate the order-preserving transformation semigroup $$OF(X, Y) = \{\alpha \in OT(X, Y) : X\alpha = Y\alpha\}.$$ Here, we characterize when an element of OF(X, Y) is regular and describe Green’s relations in OF(X, Y). Moreover, we give a simpler description of Green’s relations, characterize the ideals of OF(X, Y) when Y is a finite subset of X, and apply these results to prove that OF(X, Y) is idempotent generated.