Types of polynomial completeness for expanded groups

Description

From results of Maurer, Rhodes, and Fr\"ohlich, we know that every function on a finite simple non-abelian group is a polynomial function; these groups are called \emph{polynomially complete}. Later, it was studied when every congruence preserving function on an algebra is a polynomial function; such algebras were called \emph{affine complete}. In 2001, P.\ Idziak and K.\ S\l omczy\'nska introduced the concept of \emph{polynomial richness}. In general, it seems hard to characterize when a single algebra is affine complete or polynomially rich. Characterizations of affine complete algebras have been given for abelian groups (N\"obauer, Kaarli) and for finite algebras with a Mal'cev polynomial that have a distributive congruence lattice (Hagemann, Herrmann, Kaarli). It is not known if there is an algorithm that decides whether a given finite algebra (of finite type, with given operation tables for all operations) is affine complete. We will investigate finite modular lattices that satisfy a condition that is more general than distributivity. Based on work by Idziak and S\l omczy{\'n}ska, we can characterize affine complete and polynomially rich algebras among those finite algebras that have a group operation among their binary polynomial functions, and whose congruence lattice satisfies this condition on the congruence lattice. This is joint work with Neboj\v{s}a Mudrinski (Novi Sad).

Period	11 Feb 2006
Event title	71st Workshop on General Algebra (AAA71) together with 21st Conference for Young Algebraists (CYA21)
Event type	Conference
Location	PolandShow on map