We develop the theory of the higher commutator for Taylor varieties. A new higher commutator operation called the hypercommutator is defined using a type of invariant relation called a higher dimensional congruence. The hypercommutator is shown to be symmetric and satisfy an inequality relating nested terms. For a Taylor algebra the term condition higher commutator and the hypercommutator are equal when evaluated at a constant tuple, and it follows that every supernilpotent Taylor algebra is nilpotent. We end with a characterization of congruence meet-semidistributive varieties in terms of the neutrality of the higher commutator.

中文翻译：

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超幂零泰勒代数是幂零的

我们开发了泰勒簇的更高换向子理论。使用一种称为高维同余的不变关系定义了一种称为超交换子的新的更高交换子操作。超交换子被证明是对称的，并且满足与嵌套项相关的不等式。对于泰勒代数，条件更高交换子和超交换子在常数元组上计算时是相等的，因此每个超幂零泰勒代数都是幂零的。我们以较高交换子的中性方面的同余满足半分配变体的特征结束。