Algebra Seminar talk

Erhard Aichinger
Polynomials and Structure of Universal Algebras

In this talk, we will present some results that link the structure of a universal algebra with its clone of polynomial functions. It has recently been proved that for every finite algebra with a Mal'cev term, the clone of polynomial operations and the clone of term operations are both finitely related. This establishes that up to isomorphism and term equivalence, there are only countably many finite algebras with a Mal'cev term; in group theory this yields that for every group G, there exists a subgroup H of some finite power G^k such that for all n, all subgroups of G^n can - in a certain way - be constructed from H.

We will compare two concepts of nilpotence for expansions of groups. Starting from the well-known fact that every finite nilpotent group is a direct product of p-groups and Kearnes's generalization to finite nilpotent algebras in congruence modular varieties, we present a decomposition result for certain nilpotent infinite expanded groups.