Regularity Conditions on Order-Preserving Transformation Semigroups

In 1999, Marques-Smith and Sullivan generalised the description of Green's relations on P(X), the semigroup consisting of all partial transformations of a set X, and used it to determine the ideals of the subsemigroup generated by all nilpotents in P(X). In 2000, Kemprasit and Changphas proved that the semigroup OP(X) of order-preserving partial transformations of any chain X is regular. In this paper, we extend the earlier work on Green's relations to OP(X) where X is a countable chain with a least and a greatest element. We then describe all ideals of OP(X) for such X and determine when these ideals satisfy well-known regularity conditions. We also consider analogous problems for order-preserving total transformations.