Morita equivalence of partially ordered monoids

Two partially ordered monoids (pomonoids) are called Morita equivalent if the categories of ordered acts over them (also called S-posets) are poequivalent. We find elementary necessary and sufficient conditions for two pomonoids to be Morita equivalent. It turns out that Morita equivalent pomonoids share a number of properties. For example the congruence (or ideal) lattices of two Morita equivalent pomonoids are isomorphic.