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Morita equivalence of semigroups with local units. (English) Zbl 1229.20060
Let $S$ be a semigroup. A left $S$-act $X$ is called `closed’ if the map $\mu_X\colon S\otimes X\to X$ given by $\mu_X(s\otimes x)=sx$ is surjective and injective. The full subcategory (of all left $S$-acts and all $S$-homomorphisms) of all closed left $S$-acts is denoted by $S$-$\bold{FAct}$. Two semigroups $S$ and $T$ are called `Morita equivalent’ if the categories $S$-$\bold{FAct}$ and $T$-$\bold{FAct}$ are equivalent. A semigroup $S$ is called an `enlargement’ of its subsemigroup $U$ if $S=SUS$ and $U=USU$. A `consolidation’ of a strongly connected category $C$ is a map $q\colon\text{Obj}(C)\times\text{Obj}(C)\to\text{Mor}(C)$, $q(A,B)\colon B\to A$, such that $q(A,A)=1_A$. A small category $C$ equipped with a consolidation $q$ can be made into a semigroup $C^q$ by defining $x\circ y=xq(A,B)y$ where $x$ has domain $A$ and $y$ has codomain $B$. It is proved that semigroups with local units $S$ and $T$ are Morita equivalent if and only if one of the following conditions is satisfied: 1) the Cauchy completions $C(S)$ and $C(T)$ are equivalent; 2) $S$ and $T$ have a joint enlargement which can be chosen to be regular if $S$ and $T$ are both regular; 3) there is a unitary Morita context $(S,T,P,Q,\langle-,-\rangle,[-,-])$ with surjective mappings; 4) there is a consolidation $q$ on $C(S)$ and a local isomorphism $C(S)^q\to T$. Semigroups with local units Morita equivalent to a semigroup satisfying certain additional conditions (for example, to be a group, inverse semigroup, semilattice or orthodox semigroup) are described as well.

20M10General structure theory of semigroups
20M50Connections of semigroups with homological algebra and category theory
18B40Groupoids, semigroupoids, semigroups, groups (viewed as categories)
20M17Regular semigroups
Full Text: DOI
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