Clones of Compatible Operations on Rings Z_{p^k}}

Abstract

We investigate the lattice I(n) of clones on the ring Z n between the clone of polynomial functions and the clone of congruence preserving functions. The crucial case is when n is a prime power. For a prime p, the lattice I(p) is trivial and I(p 2 ) is known to be a 2-element lattice. We provide a description of I(p 3 ). To achieve this result, we prove a reduction theorem, which says that I(p k ) is isomorphic to a certain interval in the lattice of clones on Z p k−1 .