Algebra Seminar talk

Erhard Aichinger
Describing polynomial functions of universal algebras

From results of Fröhlich, Maurer, and Rhodes we know that every function on a finite simple non-abelian group is a polynomial function; these groups are called

polynomially complete. Later, it was studied when

every congruence preserving function on an algebra is a polynomial function; such algebras were called affine complete. We will start from the following questions:

  1. Is there an algorithm that decides whether a given

    finite algebra is affine complete?

  2. Can the polynomial functions of a finite algebra be

    described as those that preserve a certain finite set of relations?

  3. We call two finite algebras polynomially equivalent if

    they have the same set of polynomial functions. How many nonequivalent finite algebras do exist? And how many of those have a Malcev term?

We cannot answer any of these questions completely, but we will present several new results (by Juergen Ecker, Peter Mayr, Nebojsa Mudrinski, and the speaker) that are motivated by these questions.