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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 ${\mu }_{X}:S\otimes X\to X$ given by ${\mu }_{X}\left(s\otimes x\right)=sx$ 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 $S$-$\mathrm{𝐅𝐀𝐜𝐭}$. Two semigroups $S$ and $T$ are called ‘Morita equivalent’ if the categories $S$-$\mathrm{𝐅𝐀𝐜𝐭}$ and $T$-$\mathrm{𝐅𝐀𝐜𝐭}$ 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 $q:\text{Obj}\left(C\right)×\text{Obj}\left(C\right)\to \text{Mor}\left(C\right)$, $q\left(A,B\right):B\to A$, such that $q\left(A,A\right)={1}_{A}$. A small category $C$ equipped with a consolidation $q$ can be made into a semigroup ${C}^{q}$ by defining $x\circ y=xq\left(A,B\right)y$ 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 $C\left(S\right)$ and $C\left(T\right)$ 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 $\left(S,T,P,Q,〈-,-〉,\left[-,-\right]\right)$ with surjective mappings; 4) there is a consolidation $q$ on $C\left(S\right)$ and a local isomorphism $C{\left(S\right)}^{q}\to T$.

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