Strong morita equivalence and a generalisation of the rees theorem

In our earlier work we associated a natural category to a semigroup with local units and proved semigroup analogues of the celebrated Morita theorems for rings. In this article we use the notion of a Morita context to define an equivalence relation on a far wider class of semigroups. We give a semigroup analogue of Morita I and show that Morita equivalent semigroups can be constructed with ease. Our construction is based on the classical Rees theorem and a generalisation of this theorem by Hotzel. We then use properties of equivalent categories to deduce properties of this construction. Finally, we give a new generalisation of the Rees theorem and applications of this generalisation. (C) 1996 Academic Press, Inc.