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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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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Additional Information

Andrew Moorhead
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37212
Email: apmoorhead@gmail.com

DOI: https://doi.org/10.1090/tran/8251
Received by editor(s): June 19, 2019
Received by editor(s) in revised form: June 21, 2019, and June 11, 2020
Published electronically: November 12, 2020
Additional Notes: This work was supported by the National Science Foundation grant no. DMS 1500254 and the Austrian Science Fund (FWF):P29931
Article copyright: © Copyright 2020 American Mathematical Society