Clones of Compatible Operations on Rings Z_p^k

2020/07/28 by Miroslav Ploščica, Ploščica, Miroslav, Ivana Varga +1
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2007.14063

openalex publication_date 2020/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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(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(p^k) is isomorphic to a certain interval in the lattice of clones on Z_p^(k-1).

Related

• Clones containing the Mal'cev operation of ℤ_pq
• Clones from ideals
• A survey of clones on infinite sets
• Clones on infinite sets
• Large intervals in the clone lattice