Commutators and congruence preserving functions on expanded groups
by
Erhard Aichinger
Johannes Kepler Universität Linz, Austria

Much information on an algebra is contained in its lattice of congruences and the commutator operation of the congruences. Given the lattice, there are often not many ways a possible commutator operation can behave: if an algebra A in a congruence permutable variety has M₃ (the smallest modular non-distributive lattice) as its congruence lattice, then it is well-known that the commutator operation satisfies [1_A, 1_A] = 0_A, and hence the algebra is polynomially equivalent to a module over a ring. Now given Con A, we may ask ``how large'' the commutator of two congruences can become. To this end, we consider the algebra <A, Comp A >, where Comp A denotes the clone of all congruence preserving operations on A. In the case that A is a finite expanded group, we will completely determine the nonabelian and abelian prime quotients in the congruence lattice of <A, Comp  A >, and hence also the labeling of the prime quotients in the congruence lattice of <A, Comp  A> in the sense of Tame Congruence Theory. We will see that much of structure of the near-ring of zero-preserving congruence preserving functions on A can be seen from the ideal lattice of A. For example, the distributive elements of the ideal lattice of A are precisely those that are the range of an idempotent congruence preserving function of A.

In the second part of the talk, we will show why, given a finite nilpotent group, it is decidable whether the group is k-affine complete for all k in N. This part is joint work with Jürgen Ecker, Hagenberg.

Date received: April 20, 2004