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Clones of Compatible Operations on Rings โค๐k
Miroslav Ploลกฤica and Ivana Varga
We investigate the lattice ๐ผ(๐) of clones on the ring โค๐ 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 ๐ผ(๐) is trivial and ๐ผ(๐2) is known to be a 2-element lattice. We provide a description of ๐ผ(๐3). To achieve this result, we prove a reduction theorem, which says that ๐ผ(๐k) is isomorphic to a certain interval in the lattice of clones on โค๐k-1.
Keywords: Congruence, clone, polynomial