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.

翻译：克隆类是两个代数结构之间在两侧均关于其项运算复合封闭的有限元运算集合。我们猜想，对于有限模 $\mathbf A$ 与 $\mathbf B$，从 $\mathbf A$ 到 $\mathbf B$ 的克隆类为有限个当且仅当 $\mathbf A$ 与 $\mathbf B$ 的阶互素。我们对一大类模 $\mathbf A$ 证实了这一猜想。特别地，我们证明若 $\mathbf A$ 是一个有限 $k$ 维向量空间，则从 $\mathbf A$ 到任一互素模 $\mathbf B$ 的每个克隆类均可由其 $k$ 元函数生成（而 $k-1$ 元函数则不足够）。为证明此结果，我们研究了“由 $(\mathbf A,\mathbf B)$-子式一致生成”这一通用判据，并证明该判据适用于多个已有的分类结果。基于我们的分析，我们进一步证明了某些 2-幂零 Mal'cev 代数的子幂成员问题可在多项式时间内求解。