Clones of Compatible Operations on Rings ℤ𝑝k
Miroslav Ploščica and Ivana Varga
We investigate the lattice 𝐼(𝑛) of clones on the ring ℤ𝑛 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 ℤ𝑝k-1.
Keywords: Congruence, clone, polynomial