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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 ${{\Gamma }}_{M}:S\otimes S\left({}_{S}S,{}_{S}M\right)$ defined by $s\otimes t\alpha ↦\left(st\right)\alpha$ 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 $〈R,S,{}_{R}{P}_{S},{}_{S}{Q}_{R},〈,〉,\left[,\right]〉$ 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 $P\otimes {}_{S}Q$ into $R$, and $\left[,\right]$ is an $S$-$S$-morphism of $Q\otimes {}_{R}P$ into $S$ such that the following hold: 1. $〈p,q〉{p}^{\text{'}}=p\left[q,{p}^{\text{'}}\right]$ 2. $q〈p,{q}^{\text{'}}〉=\left[q,p\right]{q}^{\text{'}}$.

The author calls two semigroups $R$ and $S$ strongly Morita equivalent if there exists a unitary Morita context such that $〈,〉$ and $\left[,\right]$ 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 ${e}^{\text{'}},{f}^{\text{'}}\in S$ such that $eRf$ and ${e}^{\text{'}}S{f}^{\text{'}}$ are isomorphic semigroups.

Let $R$ be a semigroup and ${}_{R}P$, ${Q}_{R}$ be left and right $R$-acts respectively and $〈,〉:{}_{R}P×{Q}_{R}\to R$ a mapping such that $〈rp,q〉=r〈p,q〉$ and $〈p,qr〉=〈p,q〉r$ for $p\in P$, $q\in Q$, $r\in R$. Then the tensor product ${Q}_{R}\otimes {}_{R}P$ becomes a semigroup with respect to the product $\left(q\otimes p\right)\left({q}^{\text{'}}\otimes {p}^{\text{'}}\right)=q\otimes 〈p,{q}^{\text{'}}〉{p}^{\text{'}}$. This semigroup is called the Morita semigroup over $R$ defined by $〈,〉$.

It is proved that ${Q}_{R}\otimes {}_{R}P$ 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:
 20M50 Connections of semigroups with homological algebra and category theory 20M10 General structure theory of semigroups