Clones of Compatible Operations on Rings Z_(p~k)

摘要

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).