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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.

##### MSC:
 20M10 General structure theory of semigroups 20M50 Connections of semigroups with homological algebra and category theory 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories) 20M17 Regular semigroups
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