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ABSTRACT. We prove that partially ordered semigroups S and T with local units are strongly Morita equivalent if and only if there exists a surjective strict local isomorphism to T from a factorizable Rees matrix posemigroup over S. We also provide two similar descriptions which use Cauchy completions and Morita posemigroups instead.

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We introduce four extension properties (CEP, QEP, SCEP and SQEP) for ordered algebras, similar to the congruence extension property (CEP) and the strong congruence extension property of usual (unordered) algebras. All four properties turn out to have a description in terms of commutative squares or pullback diagrams. We then use these categorical descriptions to prove an ordered analogue of the well-known relation TP = AP + CEP, namely that a variety of ordered algebras has the ordered transferability property if and only if it has the ordered amalgamation property and QEP.