Thesis supervisor: Kalle Kaarli, professor emeritus, University of Tartu
Opponents: Erkko Lehtonen, D. Sc. Tech., University of Lisbon, Portugal
Peeter Puusemp, professor, Tallinn University of Technology
Algebraic structures can be similar to each other in various ways. For example, they can be identical, isomorphic, term equivalent, weakly isomorphic. For rings, a well-known notion is Morita equivalence. In the thesis, the categorical equivalence of algebraic structures is considered: two algebraic structures are categorically equivalent if the varieties they generate are equivalent as categories, and the equivalence maps one of these two structures to the other. Although the categorical equivalence is weaker notion than the weak isomorphism in general, it still preserves many important properties of algebraic structures, in particular, the property to be finite. The aim of the thesis is to explore the categorical equivalence and equivalent conditions to it within the classes of classical algebraic structures. The earlier results for finite groups, finite semigroups and finite fields served as the starting point of our study. Categorical equivalence of lattices, bands and finite rings are considered in the thesis. Motivated by the known result that finite semigroups are categorically equivalent if and only if they are weakly isomorphic, the term equivalence of semigroups is investigated systematically. In the final part of the thesis, the so called p-categorical equivalence is considered: two algebraic structures are p-categorically equivalent if the structures obtained from them by adding all constants to the set of basic operations are categorically equivalent. In such a way, some new interesting examples of categorical equivalence are obtained, based on both groups and lattices.