Peter Mayr

On clones on squarefree expanded groups

Is there a finite set with more than countably many clones that contain a Mal'cev operation? We will talk about an instance of this problem for clones on expanded groups. In particular we show that there are only finitely many polynomially inequivalent expansions of groups whose orders are a product of at most 3 distinct primes. This extends previous results by E. Aichinger and P. Mayr that characterize the clone of polynomial functions on any expanded group whose order is a product of 2 distinct primes and on any expanded group whose order is squarefree and whose congruences are linearly ordered. Still we do not have a proof for the full conjecture of P. M. Idziak that each squarefree group has only finitely many polynomially inequivalent expansions.