Supernilpotent Taylor algebras are nilpotent

Moorhead, Andrew

doi:10.1090/tran/8251

Supernilpotent Taylor algebras are nilpotent

Author: Andrew Moorhead
Journal: Trans. Amer. Math. Soc. 374 (2021), 1229-1276
MSC (2020): Primary 08A40; Secondary 08A05, 08B05
DOI: https://doi.org/10.1090/tran/8251
Published electronically: November 12, 2020
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Abstract: 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.

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