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Morita equivalence of inverse semigroups. (English) Zbl 1285.20061

The Morita theory of monoids was introduced as the analogue of the classical Morita theory for rings. Afterwards that theory was extended to semigroups with local units. This paper can be regarded as a generalization and completion of some results by one of the authors [see M. V. Lawson, Proc. Edinb. Math. Soc., II. Ser. 39, No. 3, 425-460 (1996; Zbl 0862.20047)].
The paper considers inverse semigroups and two inverse semigroups \(S\) and \(T\) are said to be Morita equivalent if the categories \(S\)-mod and \(T\)-mod (of unitary left sets and their left homomorphisms) are equivalent. The goal of the paper is to show how to get all inverse semigroups which are Morita equivalent to a fixed inverse semigroup \(S\). This is done by taking the maximum inverse images of the regular Res matrix semigroups over \(S\) where the sandwich matrix satisfies what the authors name McAlister conditions.

MSC:

20M18 Inverse semigroups
20M50 Connections of semigroups with homological algebra and category theory
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)

Citations:

Zbl 0862.20047
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References:

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