Morita invariants for partially ordered semigroups with local units (Q2896360)

According to this paper, we have the following: If \(S\) and \(T\) are strongly Morita equivalent po-semigroups with common local units and the order on \(S\) is either total, discrete or directed, then so is the order on \(T\). If \(S\) and \(T\) are strongly Morita equivalent po-semigroups with common two-sided weak local units, then their greatest commutative images are isomorphic po-semigroups, and if \(S\) satisfies an inequality, then \(T\) satisfies the same one. By a posemilattice the author means a commutative, idempotent semigroup with an order compatible with its multiplication, and it is shown that two po-semilattices which have common weak local units are strongly Morita equivalent if and only if they are isomorphic. Finally, for two strongly Morita equivalent po-semigroups with weak local units, the author defines a lattice isomorphism that maps a given type of ideals to the same type of ideals.