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arXiv · 2307.00076

Clonoids between modules

Peter Mayr·Patrick Wynne
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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.

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Peter Mayr, Patrick Wynne. 2023-06-30. Clonoids between modules. https://arxiv.org/abs/2307.00076

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