Conditions for strong Morita equivalence of partially ordered semigroups (Q657425)

In this paper the author proves, among others, that if the partially ordered semigroups (po-semigroups) \((S,\cdot,\leq)\) and \((T,\cdot,\leq)\) are strongly Morita equivalent, then the semigroups \((S,\cdot)\) and \((T,\cdot)\) are Morita equivalent; and that the strongly Morita equivalent po-semigroups are factorizable. Then the po-semigroups are characterized which are strongly Morita equivalent to a po-semigroup with local units, to a po-monoid or a po-group. Also characterized are the po-semigroups with weak local units which are strongly Morita equivalent to an orthodox po-semigroup, to an inverse po-monoid or to a simple (bisimple) regular po-monoid. As usual, \(E(S)\) denotes the set of idempotent elements of \(S\). According to this paper, a po-semigroup \(S\) is said to have local units (respectively weak local units) if for any \(s\in S\) there exist \(e,f\in E(S)\) (respectively \(e,f\in S\)) such that \(es=s=sf\). \(S\) is called factorizable if \(S^2=S\).