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Morita equivalence for semigroups. (English) Zbl 0840.20067
The author defines unitary left $S$-acts ${}_S M$ by $SM=M$ and considers semigroups with local units, i.e. for every $s\in S$ there exist idempotents $e_s,{}_sf\in S$ such that $e_s s=s= s_s f$. Parallel with {\it U. Knauer} [Semigroup Forum 3, 359-370 (1972; Zbl 0231.18013)] he develops the necessary tools to investigate Morita equivalence of two semigroups $S$ and $R$ with local units. The main difference to the case where $S$ and $R$ are monoids arises with the tensor product $S_S\otimes {}_S M$ which now is no longer isomorphic with ${}_S M$. Now ${}_S M$ is called a fixed object of the category of unitary left $S$-acts $US\text{-}\bold{Act}$ (with respect to the functor $S_S \otimes_S \Hom ({}_S S_{S,S}-)$) if $S_S\otimes {}_S M\cong {}_S M$, i.e. if the homomorphism $$\Gamma_M: S_S \otimes_S \Hom ({}_S S_S, {}_S M)\to {}_S M, \qquad (s\otimes \Phi) \mapsto {}_S\Phi$$ is an isomorphism (cf. Lemma 8.2). The full subcategory of $S$-Act (and of $US\text{-}\bold{Act}$) of those fixed objects is denoted by $FS\text{-}\bold{Act}$ and now the author calls $S$ and $R$ Morita equivalent if $FS\text{-}\bold{Act}$ and $FR\text{-}\bold{Act}$ are equivalent categories. Main Lemma: (4.8). If $S$ is a semigroup with local units and ${}_S M\in U\text{-}\bold{Act}$, then $S_S\otimes {}_S M\in FS\text{-}\bold{Act}$. Now the first basic result reads as follows: Theorem 6.1. Let $R$ and $S$ be equivalent semigroups via inverse equivalences $G: \bold{FR}\text{-}\bold{Act}\to \bold{FS}\text{-}\bold{Act}$ and $H:\bold{FS}\text{-}\bold{Act}\to\bold{FR}\text{-}\bold{Act}$. Set $P= H({}_S S)$ and $Q= G({}_R R)$. Then $P$ and $Q$ are unitary biacts ${}_R P_S$ and ${}_S Q_R$ respectively such that (1) ${}_R P$ and ${}_S Q$ are generators for $\bold{FR}\text{-}\bold{Act}$ and $\bold{FS}\text{-}\bold{Act}$ respectively; (2) $R\cong R\otimes {}_R\text{End}_S Q$, $S\cong S\otimes{}_S\text{End}_R P$ as semigroups. (3) $G\approx S \otimes {}_S\Hom_R (P,-)$, $H\approx R\otimes {}_R\Hom_S (Q,-)$. (4) ${}_S Q_R \cong S\otimes {}_S\Hom_R (P,R)$, ${}_R P_S\cong R\otimes {}_R\Hom_S (Q,S)$. Now the author proves: Theorem 9.1. A semigroup $S$ with local units is Morita equivalent to a monoid if and only if there exists $e^2= e\in S$ such that $S= SeS$. If this is the case, then $S$ is Morita equivalent to the monoid $eSe$. This immediately implies the known result for Morita equivalent monoids. As application the author gives a new proof of the Rees theorem characterizing completely 0-simple semigroups as Rees matrix semigroups. Moreover, he proves the following using a description of Rees matrix semigroups by {\it E. Hotzel} [Colloq. Math. Soc. János Bolyai 20, 247-275 (1979; Zbl 0409.20046)]. Theorem 9.8. A regular semigroup $S$ with zero is completely 0-simple if and only if $S$ is Morita equivalent to a group $G$ with zero. Theorem 9.11. A regular semigroup $S$ with zero is bisimple if and only if $S$ is Morita equivalent to a regular bisimple monoid with zero, with the Corollary 9.12. A regular semigroup $S$ is bisimple if and only if $S$ is Morita equivalent to a regular bisimple monoid. Now these three results can be used as examples for Theorem 9.1.

20M50Connections of semigroups with homological algebra and category theory