Clonoids over vector spaces

TL;DR

This paper investigates the structure and classification of clonoids between finite modules, confirming a conjecture about their finiteness under certain conditions, and explores implications for the subpower membership problem in algebra.

Contribution

It proves a conjecture relating the finiteness of clonoids to coprimality of modules and introduces a new criterion for uniform generation, with applications to Mal'cev algebras.

Findings

Finitely many clonoids exist between modules of coprime order.

Clonoids from a finite vector space are generated by their k-ary functions.

Subpower membership problem for certain Mal'cev algebras is solvable in polynomial time.

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.