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Morita equivalence for factorisable semigroups. (English) Zbl 0991.20046
Let $S$ be a semigroup with set of idempotents $E=E(S)$. 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, $US\text{-Act}=\{_SM\in S\text{-Act}\mid SM=M\}$. $US\text{-FAct}=\{M\in US\text{-Act}\mid\zeta_m=\varepsilon\}$, where $\zeta_M=\{(m_1,m_2)\mid sm_1=sm_2,\forall s\in S\}$ is an $S$-congruence on $_SM$, and $\zeta_S$ a two-sided congruence on $S$. The authors prove the following theorems: Let $R,S$ be factorisable semigroups. Let\break $UR\text{-FAct}\mathop{\rightleftharpoons}\limits^F_G US\text{-FAct}$ be a category inverse equivalence, $_RM=G(_SS')$ and $_SN=F(_RR')$. Then $_RM_S$ and $_SN_R$ are unitary biacts such that (i) $_RM$ and $_SN$ are generators in $UR$-FAct and $US$-FAct respectively; (ii) $F\approx S\Hom_R(_RM,-)$ and $G\approx R\Hom_S(_SN,-)$, where the functors are defined in the obvious way and “$\approx$” means the natural isomorphism; (iii) $S'\cong S\Hom_R({_RM},{_RM})$ and $R'\cong R\Hom_S({_SN},{_SN})$ as semigroups; (iv) $_RM_S\cong R\Hom_S(_SN_R,{_SS'})$ and $_SN_R\cong S\Hom_R(_RM_S,{_RR'})$ as biacts; (v) if we identify $_SN_R$ with $S\Hom_R(_RM_S,{_RR'})$, identify $S'$ with a right ideal of $\Hom_R(_RM,{_RM})\cong\Hom_S(_SS',{_SS'})$ and define $$\langle\ \rangle\colon M\otimes_{S'}N\to R',\ m\otimes n\mapsto\langle m,n\rangle=(m)n\text{ and }\lceil\ \rceil\colon N\otimes_{R'}M\to S',\ n\otimes m\mapsto\lceil n,m\rceil,$$ where $\lceil n,m\rceil\colon{_RM}\to{_RM}$, $x\mapsto(x)n\cdot m$, then $(R',S',{_{R'}M_{S'}},{_{S'}M_{R'}},\langle\ \rangle,\lceil\ \rceil)$ is a unitary Morita context with $\langle\ \rangle$ and $\lceil\ \rceil$ surjective. Let $R,S$ be factorisable semigroups. If there exists a unitary Morita context $(R,S,{_RP_S},{_SQ_R},\langle\ \rangle,\lceil\ \rceil)$ with $\langle\ \rangle$ and $\lceil\ \rceil$ surjective, then we have the following category inverse equivalence $UR\text{-FAct}\mathop{\rightleftharpoons}\limits^F_G US\text{-FAct}$, 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 $(R',S',{_{R'}P_{S'}},{_{S'}Q_{R'}},\langle\ \rangle,\lceil\ \rceil)$ with $\langle\ \rangle$ and $\lceil\ \rceil$ surjective. Moreover, if this is the case, then we have the following category inverse equivalence: $UR\text{-FAct}\mathop{\rightleftharpoons}\limits^F_G US\text{-FAct}$, where $F=S\Hom_R(_RP,-)$ and $G=R\Hom_S(_SQ,-)$. If there exists a unitary Morita context $(R,S,{_RP_S},{_SQ_R},\langle\ \rangle,\lceil\ \rceil)$ with $\langle\ \rangle$ and $\lceil\ \rceil$ surjective, then $R$ and $S$ are said to be strongly Morita equivalent. Define the category $S\text{-FxAct}=\{M\in US\text{-Act}\mid\Gamma_M'$ is an $S$-isomorphism$\}$ where $\Gamma_M'\colon S\otimes S\Hom_S(S,M)\to M$, $s\otimes t\cdot\phi\mapsto(st)\phi$. 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:
20M50Connections of semigroups with homological algebra and category theory
20M30Representation of semigroups; actions of semigroups on sets
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References:
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