Clonoids between modules

TL;DR

This paper studies sets of functions between finite modules that are closed under certain algebraic operations, showing finiteness under specific conditions and infinite chains otherwise, with implications for algebraic expansions.

Contribution

It establishes conditions for finiteness and infinitude of clonoids between finite modules, linking algebraic properties to the structure of these function sets.

Findings

Finite number of clonoids when modules have coprime order and distributive congruence lattice.

Existence of infinite ascending chains of clonoids without coprimality.

Any extension of modules under these conditions has infinitely many expansions.

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.