Doctoral defence: Alvin Lepik “On Morita equivalence of semigroups”

On 13 June at 15:30, Alvin Lepik will defend his doctoral thesis “On Morita equivalence of semigroups” to obtain the degree of Doctor of Philosophy (in Mathematics).

Supervisor:
Prof. Valdis Laan, University of Tartu

Opponents:
Prof. Mark Verus Lawson, Heriot-Watt University (United Kingdom)
Prof. Benjamin Steinberg, City College of New York (United States of America)

Summary
Two semigroups are regarded to be one and the same if they are isomorphic. Up to isomorphism, however, there are far too many semigroups for it to be reasonable to try and characterise them. Two semigroups are said to be Morita equivalent if the categories of all firm right acts over them are equivalent. Such a relation is a significantly weaker equivalence relation on the class of all semigroups than isomorphism. The purpose of this thesis is to study Morita equivalence of semigroups in terms of algebraic constructions used by other authors to describe Morita equivalence for various subclasses of semigroups. The objectives are to describe strong Morita equivalence classes of some well known semigroups like groups or monoids and also determine Morita invariants - these are properties that are shared by all semigroups in the same Morita equivalence class. Specifically, we shall show among other things that a factorisable semigroup is Morita equivalent to a monoid if and only if it is an enlargement of that monoid. Consequently, the enlargements of a group are precisely Rees matrix semigroups over that group and complete simplicity is a Morita invariant for factorisable semigroups. Morita equivalence of two factorisable semigroups occurs if and only if the semigroups are connected by a unitary surjective Morita context. Two chapters are reserved for the study of some morphisms and lattices induced by a Morita context. We shall obtain a description for the Morita equivalence of firm semigroups as well as show that Morita equivalent semigroups with common weak local units must have isomorphic lattices of compatible relations. The final chapter is devoted to the perfection for semigroups. We generalise many known descriptions of perfect monoids to the case of factorisable semigroups and add one more to the list. Our results allow us to conclude that the class of perfect semigroups contains all nilpotent semigroups, all completely (0-)simple semigroups and that perfection is a Morita invariant for factorisable semigroups.

Defence can be also followed in Zoom (Meeting ID: 913 0979 3632, Passcode: Lepik).