Algebra Seminar talk
Do the functions that preserve the higher commutators of a Mal'cev algebra form a clone?
Higher commutators, which were introduced by A. Bulatov, are a useful tool in congruence permutable varieties: using these higher commutators we can prove that there are at most countably many non-equivalent polynomial extensions of a finite Mal'cev algebra whose congruence lattice is of height at most 2.
For a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, we are able to show that the property of affine completeness is decidable. Moreover, the polynomial equivalence problem has polynomial complexity in the length of the input terms.
In Mal'cev algebras, a lot of properties of binary commutators can be generalized to higher commutators in a natural way. We know that all functions that preserve congruences and binary commutators of a given Mal'cev algebra form a clone. Now, one can ask the same for higher commutators. One partial result will be presented.
This is joint work with Erhard Aichinger (Linz, Austria).