Regular Elements of Some Order-Preserving Transformation Semigroups

Let X be a chain and OT(X) the full order-preserving transformation semigroup on X. In this paper, we give a necessary and sufficient condition for an element of OT(X) to be regular. For ∅ � Y ⊆ X, we may count the order-preserving transformation semigroup OT(X, Y )= {α ∈ OT(X) | ran α ⊆ Y } as a generalization of OT(X). In addition, we show that an element α ∈ OT(X, Y ) is regular in OT(X, Y ) if and only if ran α = Yα and α is regular in OT(X).