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Morita equivalence for semigroups. (English) Zbl 0840.20067

The author defines unitary left S-acts S M by SM=M and considers semigroups with local units, i.e. for every sS there exist idempotents e s , s fS such that e s s=s=s s f. Parallel with U. Knauer [Semigroup Forum 3, 359-370 (1972; Zbl 0231.18013)] he develops the necessary tools to investigate Morita equivalence of two semigroups S and R with local units.

The main difference to the case where S and R are monoids arises with the tensor product S S S M which now is no longer isomorphic with S M. Now S M is called a fixed object of the category of unitary left S-acts US-𝐀𝐜𝐭 (with respect to the functor S S S Hom( S S S,S -)) if S S S M S M, i.e. if the homomorphism

Γ M :S S S Hom( S S S , S M) S M,(sΦ) S Φ

is an isomorphism (cf. Lemma 8.2). The full subcategory of S-Act (and of US-𝐀𝐜𝐭) of those fixed objects is denoted by FS-𝐀𝐜𝐭 and now the author calls S and R Morita equivalent if FS-𝐀𝐜𝐭 and FR-𝐀𝐜𝐭 are equivalent categories. Main Lemma: (4.8). If S is a semigroup with local units and S MU-𝐀𝐜𝐭, then S S S MFS-𝐀𝐜𝐭.

Now the first basic result reads as follows: Theorem 6.1. Let R and S be equivalent semigroups via inverse equivalences G:𝐅𝐑-𝐀𝐜𝐭𝐅𝐒-𝐀𝐜𝐭 and H:𝐅𝐒-𝐀𝐜𝐭𝐅𝐑-𝐀𝐜𝐭. Set P=H( S S) and Q=G( R R). Then P and Q are unitary biacts R P S and S Q R respectively such that (1) R P and S Q are generators for 𝐅𝐑-𝐀𝐜𝐭 and 𝐅𝐒-𝐀𝐜𝐭 respectively; (2) RR R End S Q, SS S End R P as semigroups. (3) GS S Hom R (P,-), HR R Hom S (Q,-). (4) S Q R S S Hom R (P,R), R P S R R Hom S (Q,S).

Now the author proves: Theorem 9.1. A semigroup S with local units is Morita equivalent to a monoid if and only if there exists e 2 =eS such that S=SeS. If this is the case, then S is Morita equivalent to the monoid eSe.

This immediately implies the known result for Morita equivalent monoids. As application the author gives a new proof of the Rees theorem characterizing completely 0-simple semigroups as Rees matrix semigroups. Moreover, he proves the following using a description of Rees matrix semigroups by E. Hotzel [Colloq. Math. Soc. János Bolyai 20, 247-275 (1979; Zbl 0409.20046)].

Theorem 9.8. A regular semigroup S with zero is completely 0-simple if and only if S is Morita equivalent to a group G with zero. Theorem 9.11. A regular semigroup S with zero is bisimple if and only if S is Morita equivalent to a regular bisimple monoid with zero, with the

Corollary 9.12. A regular semigroup S is bisimple if and only if S is Morita equivalent to a regular bisimple monoid. Now these three results can be used as examples for Theorem 9.1.


MSC:
20M50Connections of semigroups with homological algebra and category theory