Inverse Ordered Semigroups

An element x of an ordered semigroup ( S, center dot, <= ) is called an inverse element of alpha is an element of S if alpha <= x alpha x and x <= x alpha x. An inverse ordered semigroup is an ordered semigroup S for which every element of S possesses an inverse element and the inverse elements of any element of S are in the same connected component of the Hasse diagram of its order relation. We present properties satisfied by inverse ordered semigroups and characterize inverse ordered semigroups based on the notion of regularity of ordered semigroups, their set of idempotents and Green's relations. Also we prove that the inversibility of ordered semigroups is preserved by homomorphisms.