Supernilpotent Taylor algebras are nilpotent

2019/06/21 by Moorhead, Andrew · 2 citations
#08A40 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1906.09163

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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• The class of congruence meet semidistributive varieties is not strong Maltsev
• Solvable, nilpotent and supernilpotent semigroups with completely simple ideal and monoids

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