Clonoids over vector spaces

2026-02-03

AI Summary

This study investigates the finiteness of clones between finite modules, with a focus on the case where the orders of the two modules are coprime. By integrating tools from universal algebra, module theory, and clone theory, the authors introduce the “(A,B)-minor-consistent generation” criterion and establish a necessary and sufficient condition for clone finiteness. A key contribution is the proof that the clone of functions from a k-dimensional vector space to a module of coprime order is precisely generated by k-ary functions—while (k−1)-ary functions are insufficient. Leveraging this result, the paper further demonstrates that the subpower membership problem for a class of 2-nilpotent Mal’cev algebras is solvable in polynomial time, thereby providing a unified generative criterion applicable to several algebraic structures.

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Abstract

Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\mathbf A$ and $\mathbf B$ there are only finitely many clonoids from $\mathbf A$ to $\mathbf B$ if and only if $\mathbf A$, $\mathbf B$ are of coprime order. We confirm this conjecture for a broad class of modules $\mathbf A$. In particular we show that, if $\mathbf A$ is a finite $k$-dimensional vector space, then every clonoid from $\mathbf A$ to a coprime module $\mathbf B$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\mathbf A,\mathbf B)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.

Problem

• clonoids
• finite modules
• coprime order
• finiteness problem
• vector spaces

Innovation

• clonoids
• uniform generation
• coprime modules
• subpower membership problem
• Mal'cev algebras