We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most $N$, where the number $N$ depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on $N$ that, in the special case of algebras with more than $\binom{|A|}2$ basic operations, improves an earlier result of K. Kearnes and A. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a $k$-ary near unanimity operation if and only if it contains a $k$-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.

中文翻译：

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有限代数中立方项的存在

我们研究具有有限多个基本运算的给定有限代数是否包含立方项的问题；我们给出了结构和算法结果。我们证明，如果这样的代数有一个立方项，那么它有一个维度最多为 $N$ 的立方项，其中 $N$ 的数量取决于代数基本运算的元数和基本集合的大小。对于有限幂等代数，我们在 $N$ 上给出了一个紧边界，在具有超过 $\binom{|A|}2$ 基本运算的代数的特殊情况下，改进了 K. Kearnes 和 A. Szendrei 的早期结果。在算法方面，我们表明决定立方项的存在对于幂等代数是在 P 中，在一般情况下是在 EXPTIME 中。