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Clones of Compatible Operations on Rings ℀𝑝k
Miroslav Ploőčica and Ivana Varga

We investigate the lattice 𝐼(𝑛) of clones on the ring ℀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(𝑝2) is known to be a 2-element lattice. We provide a description of 𝐼(𝑝3). To achieve this result, we prove a reduction theorem, which says that 𝐼(𝑝k) is isomorphic to a certain interval in the lattice of clones on ℀𝑝k-1.

Keywords: Congruence, clone, polynomial

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