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Morita equivalence for factorisable semigroups. (English) Zbl 0991.20046

Let $S$ be a semigroup with set of idempotents $E=E\left(S\right)$. Then $S$ is called factorisable if $S={S}^{2}$; $S$ is called sandwich if $S=SES$; $S$ is called local units if $S=ESE$.

$S$-Act denotes the category of left $S$-acts, $US\text{-Act}={\left\{}_{S}M\in S\text{-Act}\mid SM=M\right\}$. $US\text{-FAct}=\left\{M\in US\text{-Act}\mid {\zeta }_{m}=\epsilon \right\}$, where ${\zeta }_{M}=\left\{\left({m}_{1},{m}_{2}\right)\mid s{m}_{1}=s{m}_{2},\forall s\in S\right\}$ is an $S$-congruence on ${}_{S}M$, and ${\zeta }_{S}$ a two-sided congruence on $S$.

The authors prove the following theorems: Let $R,S$ be factorisable semigroups. Let$UR\text{-FAct}\underset{G}{\overset{F}{⇌}}US\text{-FAct}$ be a category inverse equivalence, ${}_{R}M=G{\left(}_{S}{S}^{\text{'}}\right)$ and ${}_{S}N=F{\left(}_{R}{R}^{\text{'}}\right)$. Then ${}_{R}{M}_{S}$ and ${}_{S}{N}_{R}$ are unitary biacts such that (i) ${}_{R}M$ and ${}_{S}N$ are generators in $UR$-FAct and $US$-FAct respectively; (ii) $F\approx S{\text{Hom}}_{R}{\left(}_{R}M,-\right)$ and $G\approx R{\text{Hom}}_{S}{\left(}_{S}N,-\right)$, where the functors are defined in the obvious way and “$\approx$” means the natural isomorphism; (iii) ${S}^{\text{'}}\cong S{\text{Hom}}_{R}\left({}_{R}M,{}_{R}M\right)$ and ${R}^{\text{'}}\cong R{\text{Hom}}_{S}\left({}_{S}N,{}_{S}N\right)$ as semigroups; (iv) ${}_{R}{M}_{S}\cong R{\text{Hom}}_{S}{\left(}_{S}{N}_{R},{}_{S}{S}^{\text{'}}\right)$ and ${}_{S}{N}_{R}\cong S{\text{Hom}}_{R}{\left(}_{R}{M}_{S},{}_{R}{R}^{\text{'}}\right)$ as biacts; (v) if we identify ${}_{S}{N}_{R}$ with $S{\text{Hom}}_{R}{\left(}_{R}{M}_{S},{}_{R}{R}^{\text{'}}\right)$, identify ${S}^{\text{'}}$ with a right ideal of ${\text{Hom}}_{R}{\left(}_{R}M,{}_{R}M\right)\cong {\text{Hom}}_{S}{\left(}_{S}{S}^{\text{'}},{}_{S}{S}^{\text{'}}\right)$ and define

$〈\phantom{\rule{4pt}{0ex}}〉:M{\otimes }_{{S}^{\text{'}}}N\to {R}^{\text{'}},\phantom{\rule{4pt}{0ex}}m\otimes n↦〈m,n〉=\left(m\right)n\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}⌈\phantom{\rule{4pt}{0ex}}⌉:N{\otimes }_{{R}^{\text{'}}}M\to {S}^{\text{'}},\phantom{\rule{4pt}{0ex}}n\otimes m↦⌈n,m⌉,$

where $⌈n,m⌉:{}_{R}M\to {}_{R}M$, $x↦\left(x\right)n·m$, then $\left({R}^{\text{'}},{S}^{\text{'}},{}_{{R}^{\text{'}}}{M}_{{S}^{\text{'}}},{}_{{S}^{\text{'}}}{M}_{{R}^{\text{'}}},〈\phantom{\rule{4pt}{0ex}}〉,⌈\phantom{\rule{4pt}{0ex}}⌉\right)$ is a unitary Morita context with $〈\phantom{\rule{4pt}{0ex}}〉$ and $⌈\phantom{\rule{4pt}{0ex}}⌉$ surjective. Let $R,S$ be factorisable semigroups. If there exists a unitary Morita context $\left(R,S,{}_{R}{P}_{S},{}_{S}{Q}_{R},〈\phantom{\rule{4pt}{0ex}}〉,⌈\phantom{\rule{4pt}{0ex}}⌉\right)$ with $〈\phantom{\rule{4pt}{0ex}}〉$ and $⌈\phantom{\rule{4pt}{0ex}}⌉$ surjective, then we have the following category inverse equivalence $UR\text{-FAct}\underset{G}{\overset{F}{⇌}}US\text{-FAct}$, where $F=Q{\otimes }_{R}-/{\zeta }_{Q{\otimes }_{R}-}$ and $G=P{\otimes }_{S}-/{\zeta }_{P{\otimes }_{S}-}$.

Let $R,S$ be factorisable semigroups. Then the category $UR$-FAct is equivalent to the category $US$-FAct if and only if there exists a unitary Morita context $\left({R}^{\text{'}},{S}^{\text{'}},{}_{{R}^{\text{'}}}{P}_{{S}^{\text{'}}},{}_{{S}^{\text{'}}}{Q}_{{R}^{\text{'}}},〈\phantom{\rule{4pt}{0ex}}〉,⌈\phantom{\rule{4pt}{0ex}}⌉\right)$ with $〈\phantom{\rule{4pt}{0ex}}〉$ and $⌈\phantom{\rule{4pt}{0ex}}⌉$ surjective. Moreover, if this is the case, then we have the following category inverse equivalence: $UR\text{-FAct}\underset{G}{\overset{F}{⇌}}US\text{-FAct}$, where $F=S{\text{Hom}}_{R}{\left(}_{R}P,-\right)$ and $G=R{\text{Hom}}_{S}{\left(}_{S}Q,-\right)$. If there exists a unitary Morita context $\left(R,S,{}_{R}{P}_{S},{}_{S}{Q}_{R},〈\phantom{\rule{4pt}{0ex}}〉,⌈\phantom{\rule{4pt}{0ex}}⌉\right)$ with $〈\phantom{\rule{4pt}{0ex}}〉$ and $⌈\phantom{\rule{4pt}{0ex}}⌉$ surjective, then $R$ and $S$ are said to be strongly Morita equivalent.

Define the category $S\text{-FxAct}=\left\{M\in US\text{-Act}\mid {{\Gamma }}_{M}^{\text{'}}$ is an $S$-isomorphism$\right\}$ where ${{\Gamma }}_{M}^{\text{'}}:S\otimes S{\text{Hom}}_{S}\left(S,M\right)\to M$, $s\otimes t·\phi ↦\left(st\right)\phi$.

Now the authors prove: (i) If either ${\prod }_{i\in I}S\in S\text{-FxAct}$ for any index set $I$ or $S\in US$-FAct holds, then $S$ is strongly Morita equivalent to a monoid if and only if $S=SeS$ for some $e\in E$. Moreover, if this is the case, then $S$ is strongly Morita equivalent to $eSe$. (ii) If ${\prod }_{i\in I}S\in S\text{-FxAct}$ for any index set $I$, then the following statements are equivalent: (a) $S=SES$; (b) $S$ is strongly Morita equivalent to a sandwich semigroup; (c) $S$ is strongly Morita equivalent to a local units semigroup. Let $S$ be an arbitrary semigroup. Then $S$ is a completely simple semigroup if and only if ${\prod }_{i\in I}S\in S\text{-FxAct}$ for any index set $I$ and $S$ is strongly Morita equivalent to a group.

Let $S$ be a like unity semigroup. Then the following statements hold: (i) $S$ is strongly Morita equivalent to a group if and only if $S$ is a completely simple semigroup; (ii) $S$ is strongly Morita equivalent to a monoid if and only if $S=SeS$ for some $e\in E$; (iii) $S$ is strongly Morita equivalent to a local semigroup if and only if $S=SES$. Here a semigroup is called like unity if $x\in Sx\cap xS$ for any $x\in S$.

##### MSC:
 20M50 Connections of semigroups with homological algebra and category theory 20M30 Representation of semigroups; actions of semigroups on sets
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