Centrality and Internal Structures in Universal Algebra
A variety of universal algebras is Maltsev if it has a ternary term p(x,y,z) satisfying the axioms p(x,y,y)=x and p(x,x,y)=y. Accordingly, the varieties of groups, rings, Lie algebras, quasigroups, crossed modules and Heyting algebras are all examples of Maltsev varieties. We shall first present a new approach to investigate the classical property of centrality of congruence relations in these varieties. The internal notion of connector of equivalence relations is introduced, giving a better understanding of centrality in the more general context of regular Maltsev categories. In this categorical setting the important examples of Maltsev quasivarieties and of topological groups are also included. In the second part of the talk we shall compare these ideas with the recent categorical theory of central extensions developed by Janelidze and Kelly. We shall then show that in any Maltsev category with a zero object there is a functor associating a connected internal groupoid with any central extension. This general construction clarifies the well-known fact that any central extension can be considered as a crossed module.
1. J. D. H. Smith, Mal’cev Varieties, LNM 554, Springer-Verlag, 1976.
2. G. Janelidze, G. M. Kelly, Galois theory and a general notion of central extension, J. Pure Appl. Algebra, 97, 1994, 135-161.
3. D. Bourn, M. Gran, Centrality and connectors in Maltsev categories, Preprint 2001.
4. M. Gran, Algebraically central and categorically central extensions, Pré-publicações da Universidade de Coimbra, 01-02, Fevereiro 2001.
Marino Gran (Univ. Catholique Louvain, Belgium)
March 14, 2001