Finite generation and finiteness of clonoids between coprime-order abelian Mal'cev algebras

Prove that, for abelian Mal'cev algebras U and L of coprime order, every (U,L)-clonoid is finitely generated and only finitely many (U,L)-clonoids exist.

Background

A clonoid from U to L is a collection of finitary functions from powers of U to L closed under composition with term functions of U on the input side and term functions of L on the output side. The paper uses clonoids, particularly difference clonoids, to analyze term clones of central extensions and to obtain polynomial-time algorithms for selected subpower membership problems.

Known results establish finite generation and finiteness for clonoids from finite vector spaces to finite modules of coprime order, as well as for certain distributive-module settings. The conjecture proposes the corresponding statement for arbitrary finite abelian Mal'cev algebras of coprime order. The paper identifies this conjecture as potentially useful for proving tractability of SMP(L tensor U) in the general 2-nilpotent setting.