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 \(\left( {\begin{array}{c}|A|\\ 2\end{array}}\right) \) basic operations, improves an earlier result of K. Kearnes and Á. 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.

我们研究一个具有有限多个基本运算的有限代数是否包含立方项的问题。我们给出结构和算法结果。我们证明，如果这样的代数具有立方项，那么它的维数立方项最多为N，其中数N取决于代数的基本运算和基本集的大小。对于有限幂等代数，我们在N上给出一个严格的界，在特殊情况下，具有大于\（\ left（{\ begin {array} {c} | A | \\ 2 \ end {array}} \ right ）\）基本运算，改善了K. Kearnes和Á的早期结果。Szendrei。在算法方面，我们表明确定立方体项的存在在于P一般用于幂等代数和EXPTIME。由于一个代数在且仅当它包含一个k维立方项并且生成一个全等分布变型时，才包含一个k元近似一致运算，因此我们的算法还可以让我们确定一个给定的有限代数是否具有一个近似一致运算。