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Strong Morita equivalence and a generalisation of the Rees theorem. (English) Zbl 0855.20054

Continuing earlier papers the author again considers Morita equivalent semigroups, i.e. semigroups R and S such that the categories R-FxAct and S-FxAct are equivalent. Here S-FxAct consists of unitary left S-acts S M such that the canonical S-homomorphism Γ M :SS( S S, S M) defined by stα(st)α is an isomorphism and S M is called unitary if SM=M. A semigroup S which is unitary as a left S-act, i.e. fulfills S=S 2 is called factorizable.

A six-tuple R,S, R P S , S Q R ,,,[,] is said to be a unitary Morita context, where R and S are factorisable semigroups, R P S and S Q R are unitary R-S- and S-R-biacts respectively, , is an R-R-morphism of P S Q into R, and [,] is an S-S-morphism of Q R P into S such that the following hold: 1. p,qp ' =p[q,p ' ] 2. qp,q ' =[q,p]q ' .

The author calls two semigroups R and S 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 E in S such that S=SES. If R and S are sandwich semigroups which are strongly Mority equivalent, then the subcategories of principal projectives are equivalent and the cardinalities of sets of regular D-classes of R and S are equivalent. If e,fR are idempotent, then there exist idempotents e ' ,f ' S such that eRf and e ' Sf ' are isomorphic semigroups.

Let R be a semigroup and R P, Q R be left and right R-acts respectively and ,: R P×Q R R a mapping such that rp,q=rp,q and p,qr=p,qr for pP, qQ, rR. Then the tensor product Q R R P becomes a semigroup with respect to the product (qp)(q ' p ' )=qp,q ' p ' . This semigroup is called the Morita semigroup over R defined by ,.

It is proved that Q R R P is strongly Morita equivalent to R if R is factorizable, P and Q 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 [14]) 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.


MSC:
20M50Connections of semigroups with homological algebra and category theory
20M10General structure theory of semigroups