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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 ${e}_{s},{}_{s}f\in S$ such that ${e}_{s}s=s={s}_{s}f$. Parallel with 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 $US\text{-}\mathrm{𝐀𝐜𝐭}$ (with respect to the functor ${S}_{S}{\otimes }_{S}\text{Hom}\left({}_{S}{S}_{S,S}-\right)$) if ${S}_{S}\otimes {}_{S}M\cong {}_{S}M$, i.e. if the homomorphism

${{\Gamma }}_{M}:{S}_{S}{\otimes }_{S}\text{Hom}\left({}_{S}{S}_{S},{}_{S}M\right)\to {}_{S}M,\phantom{\rule{2.em}{0ex}}\left(s\otimes {\Phi }\right)↦{}_{S}{\Phi }$

is an isomorphism (cf. Lemma 8.2). The full subcategory of $S$-Act (and of $US\text{-}\mathrm{𝐀𝐜𝐭}$) of those fixed objects is denoted by $FS\text{-}\mathrm{𝐀𝐜𝐭}$ and now the author calls $S$ and $R$ Morita equivalent if $FS\text{-}\mathrm{𝐀𝐜𝐭}$ and $FR\text{-}\mathrm{𝐀𝐜𝐭}$ are equivalent categories. Main Lemma: (4.8). If $S$ is a semigroup with local units and ${}_{S}M\in U\text{-}\mathrm{𝐀𝐜𝐭}$, then ${S}_{S}\otimes {}_{S}M\in FS\text{-}\mathrm{𝐀𝐜𝐭}$.

Now the first basic result reads as follows: Theorem 6.1. Let $R$ and $S$ be equivalent semigroups via inverse equivalences $G:\mathrm{𝐅𝐑}\text{-}\mathrm{𝐀𝐜𝐭}\to \mathrm{𝐅𝐒}\text{-}\mathrm{𝐀𝐜𝐭}$ and $H:\mathrm{𝐅𝐒}\text{-}\mathrm{𝐀𝐜𝐭}\to \mathrm{𝐅𝐑}\text{-}\mathrm{𝐀𝐜𝐭}$. Set $P=H\left({}_{S}S\right)$ and $Q=G\left({}_{R}R\right)$. 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 $\mathrm{𝐅𝐑}\text{-}\mathrm{𝐀𝐜𝐭}$ and $\mathrm{𝐅𝐒}\text{-}\mathrm{𝐀𝐜𝐭}$ respectively; (2) $R\cong R\otimes {}_{R}{\text{End}}_{S}Q$, $S\cong S\otimes {}_{S}{\text{End}}_{R}P$ as semigroups. (3) $G\approx S\otimes {}_{S}{\text{Hom}}_{R}\left(P,-\right)$, $H\approx R\otimes {}_{R}{\text{Hom}}_{S}\left(Q,-\right)$. (4) ${}_{S}{Q}_{R}\cong S\otimes {}_{S}{\text{Hom}}_{R}\left(P,R\right)$, ${}_{R}{P}_{S}\cong R\otimes {}_{R}{\text{Hom}}_{S}\left(Q,S\right)$.

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 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.

##### MSC:
 20M50 Connections of semigroups with homological algebra and category theory