Continuing earlier papers the author again considers Morita equivalent semigroups, i.e. semigroups and such that the categories -FxAct and -FxAct are equivalent. Here -FxAct consists of unitary left -acts such that the canonical -homomorphism defined by is an isomorphism and is called unitary if . A semigroup which is unitary as a left -act, i.e. fulfills is called factorizable.
A six-tuple is said to be a unitary Morita context, where and are factorisable semigroups, and are unitary -- and --biacts respectively, is an --morphism of into , and is an --morphism of into such that the following hold: 1. 2. .
The author calls two semigroups and strongly Morita equivalent if there exists a unitary Morita context such that and are surjective. It is proved that strong Morita equivalence implies Morita equivalence. A semigroup is called a sandwich semigroup if there exists a set of idempotents in such that . If and are sandwich semigroups which are strongly Mority equivalent, then the subcategories of principal projectives are equivalent and the cardinalities of sets of regular -classes of and are equivalent. If are idempotent, then there exist idempotents such that and are isomorphic semigroups.
Let be a semigroup and , be left and right -acts respectively and a mapping such that and for , , . Then the tensor product becomes a semigroup with respect to the product . This semigroup is called the Morita semigroup over defined by .
It is proved that is strongly Morita equivalent to if is factorizable, and are unitary and is surjective. As an application the author shows that a completely 0-simple semigroup gives rise to a Morita context. Then using the result of a former paper of the author (cited as ) that a completely 0-simple semigroup is Morita equivalent to a group with 0, he gets a new proof of the classical Rees structure theorem.