Morita equivalence of semigroups
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Morita equivalence is a general way of classifying structures by means of their actions that is weaker than isomorphism but at the same time useful. It arose first in the study of unital rings in the 1950’s  but has since been extended to many other kinds of strucures, including classes of non- unital rings. It was first applied to semigroup theory in the 1970’s in the work of Banaschewski  and Knauer  who independently determined when two monoids were Morita equivalent. However they were unable to extend their definition to arbitrary semigroups since Banaschewski showed that Morita equivalence reduced to isomorphism. It was not until the 1990’s that Talwar [40, 41] was able to find a good definition of Morita equivalence for a class of semigroups that included all monoids but also all regular semigroups: the class of semigroups with local units. Such a semigroup is one in which each element has a left and a right idempotent identity. Talwar’s work was not developed further until the twenty-first century when a variety of mathematicians including Funk, Laan, Lawson, M´arki and Steinberg started to develop the Morita theory of semigroups in detail [9, 20, 25, 39]. Our thesis takes as its starting point Lawson’s reinterpretation of Talwar’s work. The thesis consists of three chapters. An essential ingredient in Morita theory is the notion of an equivalence of categories. For this reason, Chapter 1 of this thesis reviews all the categorical definitions needed. In Chapter 2, we describe in detail the work of Banaschewski and Knauer on the Morita theory of monoids. These two chapters contain no new work. We begin Chapter 3 by explaining why the obvious way of defining the Morita equivalence of two semigroups does not work. We then describe Lawson’s approach to Talwar’s work. This provides the foundation for our thesis. Our new contributions to the theory are contained in Sections 3.2, 3.3 and 3.4 and are based on Rees matrix semigroups. Talwar showed that the classical Rees matrix theorem for completely simple semigroups could be regarded as a Morita theorem: a semigroup is Morita equivalent to a group if and only if it is completely simple if and only if it is isomorphic to a Rees matrix semigroup over a group. This raises the question of determining what role Rees matrix semigroups play in the Morita theory of semigroups with local units. We investigate three different problems based on this idea: Section 3.2 In this section, we try to provean exact generalisation of the Rees theorem. We are interested in the case where S is Morita equivalent to T if and only if S is isomorphic to some kind of Rees matrix semigroup over T. Section 3.3 In this section, we prove that S is Morita equivalent to T if and only if S is a locally isomorphic image of a special kind of Rees matrix semigroup over T. This result was first proved by Laan and M´arki  but we give a new proof that generalizes the classical proof of the Rees theorem. Section 3.4 Finally, we solve the following problem: given an inverse semi-group S find all inverse semigroups T which are Morita equivalent to S. Our solution uses special kinds of Rees matrix semigroups over S. In this section, we also describe those semigroups which are Morita equivalent to semigroups with commuting idempotents. This builds on early work by Khan and Lawson [17, 18].