2-幂零Mal'cev代数的结构与复杂性
Structure and Complexity of 2-Nilpotent Mal'cev Algebras
浏览论文内容
中文总结 AI 辅助
该研究利用差clonoid分析同余模簇中代数中心扩张的结构，刻画2步幂零代数数量有限的条件，并证明一大类幂零Mal'cev代数的子幂隶属问题可多项式时间求解。
中文摘要 AI 辅助
我们研究同余模簇中代数的中心扩张结构。我们使用一种称为clonoid（克隆类）的多类代数对象来理解这类中心扩张的项克隆。我们构建了这类中心扩张的差clonoid，并利用它证明：固定有限集合上的2步幂零代数的数量是有限的，当且仅当该集合的阶是无平方因子的。有限代数结构$\u211d$的子幂隶属问题是指：给定输入$a_1,\dots,a_k, b \in A^n$，判断$b$是否属于由$a_1, \dots, a_k$生成的$\u211d^n$的子代数。我们证明，对于一大类幂零Mal'cev代数，子幂隶属问题可在多项式时间内求解，尤其是无平方因子阶的2步幂零Mal'cev代数。
英文摘要
We investigate the structure of central extensions for algebras in a congruence modular variety. We use a multisorted algebraic object called a clonoid to understand the term clone of such a central extension. We develop the difference clonoid of such a central extension and use it to show that the number of $2$-step nilpotent algebras on a fixed finite set is finite if and only if the set is of squarefree order. The subpower membership problem for a finite algebraic structure $\mathbb{A}$ is the problem of deciding on input $a_1,\dots,a_k, b \in A^n$, whether $b$ is in the subalgebra of $\mathbb{A}^n$ generated by $a_1, \dots, a_k$. We show that for a large class of nilpotent Mal'cev algebras the subpower membership problem is solvable in polynomial time, in particular for $2$-step nilpotent Mal'cev algebras of squarefree order.
