向向量空间的克龙体
Clonoids over vector spaces
AI总结：
本文研究了向量空间的克龙体，证明了在特定条件下克龙体的有限性，并展示了相关代数问题的多项式时间解法。
摘要：
克龙体是两个代数结构之间的一组有限元操作，它们在两边的术语操作下闭合。我们猜测，对于有限模A和B，从A到B的克龙体只有有限多个当且仅当A和B的阶数互质。我们验证了这一猜想在广泛类别的模A上成立。特别是我们证明，如果A是一个有限k维向量空间，那么从A到一个互质模B的每个克龙体都由其k元函数生成（ arity k-1 不足以）。为了证明这一结果，我们研究了“由（A,B）- minors均匀生成”的一般标准，我们证明它适用于其他已有的分类结果。基于我们的分析，我们进一步证明了某些2- nilpotent Mal'cev代数的子幂成员问题可以在多项式时间内解决。
英文摘要：
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.
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