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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 $R$-{\bf FxAct} and $S$-{\bf FxAct} are equivalent. Here $S$-{\bf FxAct} consists of unitary left $S$-acts ${_SM}$ such that the canonical $S$-homomorphism $\Gamma_M:S\otimes S({_SS},{_SM})$ defined by $s\otimes t\alpha\mapsto (st)\alpha$ is an isomorphism and ${_SM}$ 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 $\langle R,S,{_RP_S},{_SQ_R},\langle , \rangle,[, ]\rangle$ is said to be a unitary Morita context, where $R$ and $S$ are factorisable semigroups, ${_RP_S}$ and ${_SQ_R}$ are unitary $R$-$S$- and $S$-$R$-biacts respectively, $\langle , \rangle$ is an $R$-$R$-morphism of $P\otimes{_SQ}$ into $R$, and $[, ]$ is an $S$-$S$-morphism of $Q\otimes{_RP}$ into $S$ such that the following hold: 1. $\langle p,q\rangle p'=p[q,p']$ 2. $q\langle p,q'\rangle=[q,p]q'$. The author calls two semigroups $R$ and $S$ strongly Morita equivalent if there exists a unitary Morita context such that $\langle , \rangle$ 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 $e,f\in R$ are idempotent, then there exist idempotents $e',f'\in S$ such that $eRf$ and $e'Sf'$ are isomorphic semigroups. Let $R$ be a semigroup and ${_RP}$, $Q_R$ be left and right $R$-acts respectively and $\langle , \rangle:{_RP}\times Q_R\to R$ a mapping such that $\langle rp,q\rangle=r\langle p,q\rangle$ and $\langle p,qr\rangle=\langle p,q\rangle r$ for $p\in P$, $q\in Q$, $r\in R$. Then the tensor product $Q_R\otimes{_RP}$ becomes a semigroup with respect to the product $(q\otimes p)(q'\otimes p')=q\otimes\langle p,q'\rangle p'$. This semigroup is called the Morita semigroup over $R$ defined by $\langle , \rangle$. It is proved that $Q_R\otimes{_RP}$ is strongly Morita equivalent to $R$ if $R$ is factorizable, $P$ and $Q$ are unitary and $\langle , \rangle$ 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.

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