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

TL;DR

This paper explores the structure of clones of compatible operations on rings Z_{p^k}, especially for prime powers, revealing new insights into their lattice organization and providing a description of I(p^3).

Contribution

It introduces a reduction theorem linking I(p^k) to a lattice interval on Z_p^{k-1}, advancing understanding of clone structures on prime power rings.

Findings

I(p) is trivial for prime p

I(p^2) forms a 2-element lattice

I(p^3) description is provided

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).