Clonoids between modules

Abstract

Clonoids are sets of finitary functions from an algebra $\mathrm{\text{A}}$ to an algebra $\mathrm{\text{B}}$ that are closed under composition with term functions of $\mathrm{\text{A}}$ on the domain side and with term functions of $\mathrm{\text{B}}$ on the codomain side. For $\mathrm{\text{A, B}}$ (polynomially equivalent to) finite modules we show: If $\mathrm{\text{A, B}}$ have coprime order and the congruence lattice of $\mathrm{\text{A}}$ is distributive, then there are only finitely many clonoids from $\mathrm{\text{A}}$ to $\mathrm{\text{B}}$. This is proved by establishing for every natural number $k$ a particular linear equation that all $k$-ary functions from $\mathrm{\text{A}}$ to $\mathrm{\text{B}}$ satisfy. Else if $\mathrm{\text{A, B}}$ do not have coprime order, then there exist infinite ascending chains of clonoids from $\mathrm{\text{A}}$ to $\mathrm{\text{B}}$ ordered by inclusion. Consequently any extension of $\mathrm{\text{A}}$ by $\mathrm{\text{B}}$ has countably infinitely many $2$-nilpotent expansions up to term equivalence.