Representation of inverse ordered semigroups by means of symmetric inverse semigroups on a set

A partial one-to-one mapping on a set X is a mapping whose domain is a subset of X. The set of partial one-one mappings on a set X is an inverse semigroup (the symmetric inverse semigroup on X). An element x of an ordered semigroup (S, ·, ≤) is called an inverse element of α ∈ S if α ≤ xαx and x ≤ xαx. 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. In the paper, we give an analogous result to the Vagner-Preston Representation for inverse ordered semigroups by proving that for every inverse ordered semigroup S there exists an ℛ≤-monoantimorphism from S to the symmetric inverse semigroup on S.