Arithmetical affine complete varieties and inverse monoids

This paper gives a classification of arithmetical affine complete varieties of finite type up to categorical equivalence. It is proved that two such varieties are equivalent as categories if and only if their weakly diagonal generators have isomorphic monoids of bicongruences. Moreover, it is proved that the monoids appearing in this situation are precisely the inverse factorizable monoids with zero, with distributive lattice of idempotents, and satisfying a certain idempotent-unit condition (IU).

Affine Completeness of Generalised Dihedral Groups

In this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every k ∈ N the k-affine complete generalised dihedral groups. We find that the direct product of a 1-affine complete group with itself need not be 1-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for 2-affine completeness.

EVERY (k + 1)-AFFINE COMPLETE NILPOTENT GROUP OF CLASS k IS AFFINE COMPLETE

We let G be a group, and we let k be a natural number. We assume that G is nilpotent of class at most k, and that every (k + 1)-ary congruence preserving function on G is a polynomial function. We show that then every congruence preserving function on G (of any finite arity) is a polynomial function.