The author defines unitary left -acts by and considers semigroups with local units, i.e. for every there exist idempotents such that . 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 and with local units.
The main difference to the case where and are monoids arises with the tensor product which now is no longer isomorphic with . Now is called a fixed object of the category of unitary left -acts (with respect to the functor ) if , i.e. if the homomorphism
is an isomorphism (cf. Lemma 8.2). The full subcategory of -Act (and of ) of those fixed objects is denoted by and now the author calls and Morita equivalent if and are equivalent categories. Main Lemma: (4.8). If is a semigroup with local units and , then .
Now the first basic result reads as follows: Theorem 6.1. Let and be equivalent semigroups via inverse equivalences and . Set and . Then and are unitary biacts and respectively such that (1) and are generators for and respectively; (2) , as semigroups. (3) , . (4) , .
Now the author proves: Theorem 9.1. A semigroup with local units is Morita equivalent to a monoid if and only if there exists such that . If this is the case, then is Morita equivalent to the monoid .
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 with zero is completely 0-simple if and only if is Morita equivalent to a group with zero. Theorem 9.11. A regular semigroup with zero is bisimple if and only if is Morita equivalent to a regular bisimple monoid with zero, with the
Corollary 9.12. A regular semigroup is bisimple if and only if is Morita equivalent to a regular bisimple monoid. Now these three results can be used as examples for Theorem 9.1.