This paper considers the complexity of the problem of determining whether a finite idempotent algebra generates a variety which satisfies various strong Mal’tsev conditions. In particular we show that if the strong Mal’tsev condition under consideration implies the existence of an edge term, then deciding whether a finite idempotent algebra generates a variety which satisfies the Mal’tsev condition is in the complexity class NP. This extends an earlier result of Kazda, Opršal, Valeriote and Zhuk concerning minority terms to a much broader class of conditions, notably including all nontrivial linear strong Mal’tsev conditions in which every equation relates a term to a variable. The result of Kazda et al. being an application of a result of Mayr’s concerning subpower membership problems, we note that our new result may similarly be seen as an application of the results on subpower membership problems obtained by Bulatov, Mayr and Szendrei. We also show that for fixed \(k \ge 2\) there is a polynomial-time procedure which takes an idempotent algebra \({\mathbf {A}}\) and produces the operation table of a k-edge term operation of \({\mathbf {A}}\) if such a term exists or else correctly returns the answer that no such term exists, again extending an earlier result of Kazda et al.

本文考虑了确定有限幂等代数是否生成满足各种强 Mal'tsev 条件的变体的问题的复杂性。特别地，我们表明，如果考虑中的强 Mal'tsev 条件意味着边项的存在，那么确定有限幂等代数是否生成满足 Mal'tsev 条件的变体在复杂度类NP 中. 这将 Kazda、Opršal、Valeriote 和 Zhuk 关于少数项的早期结果扩展到更广泛的条件类别，特别是包括所有非平凡线性强 Mal'tsev 条件，其中每个方程都将一个项与一个变量相关联。Kazda 等人的结果。作为迈尔关于次权力隶属问题的结果的应用，我们注意到我们的新结果同样可以被视为布拉托夫、迈尔和森德雷在次权力隶属问题上的结果的应用。我们还表明，对于固定\(k \ge 2\)有一个多项式时间过程，它采用幂等代数\({\mathbf {A}}\)并产生k边项运算的运算表\({\mathbf {A}}\) 如果这样的术语存在，或者正确返回不存在这样的术语的答案，再次扩展 Kazda 等人的早期结果。