Conditions for Strong Morita equivalence of partially ordered semigroups

We investigate when a partially ordered semigroup (with various types of local units) is strongly Morita equivalent to a posemigroup from a given class of partially ordered semigroups. We are able to provide necessary and sufficient conditions for this to happen for a series of well-known classes of posemigroups. We further include a number of sufficient conditions for several classes of naturally ordered posemigroups.