Clonoids between modules

Abstract

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