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Clones of Compatible Operations on Rings โ„ค๐‘k
Miroslav Ploลกฤica and Ivana Varga

We investigate the lattice ๐ผ(๐‘›) of clones on the ring โ„ค๐‘› between the clone of polynomial functions and the clone of congruence preserving functions. The crucial case is when ๐‘› is a prime power. For a prime ๐‘, the lattice ๐ผ(๐‘) is trivial and ๐ผ(๐‘2) is known to be a 2-element lattice. We provide a description of ๐ผ(๐‘3). To achieve this result, we prove a reduction theorem, which says that ๐ผ(๐‘k) is isomorphic to a certain interval in the lattice of clones on โ„ค๐‘k-1.

Keywords: Congruence, clone, polynomial

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