On supernilpotent algebras
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We establish a characterization of supernilpotent Mal’cev algebras which generalizes the affine structure of abelian Mal’cev algebras and the recent characterization of 2-supernilpotent Mal’cev algebras. We then show that for varieties in which the two-generated free algebra is finite: (1) neutrality of the higher commutators is equivalent to congruence meet-semidistributivity, and (2) the class of varieties which interpret a Mal’cev term in every supernilpotent algebra is equivalent to the existence of a weak difference term. We then establish properties of the higher commutator in the aforementioned second class of varieties.
KeywordsHigher commutator Supernilpotent Polynomial equivalence
Mathematics Subject Classification03C05 08B05 08A30
I would like to thank Andrew Moorhead and Jakub Opršal for intriguing and enthusiastic discussions about the higher commutator during the Vanderbilt Workshop on Structure and Complexity in Universal Algebra held September 19–30, 2016 in Nashville, TN.
- 4.Bulatov, A.: On the number of finite Mal’tsev algebras. In: Contr. Gen. Alg. 13, Proceedings of the Dresden Conference 2000 (AAA 60) and the Summer School 1999, pp. 41–54. Verlag Johannes Heyn, Klagenfurt (2001)Google Scholar
- 9.Gumm, H.P.: Geometrical methods in congruence modular algebras. Mem. Amer. Math. Soc. 21(286) (1983)Google Scholar
- 14.Kearnes, K., Kiss, E.: The Shape of Congruence Lattices. Memoirs of the American Mathematical Society 222, 1046 (2013)Google Scholar
- 19.Smith, J.D.H.: Mal’cev varieties. Lecture Notes in Mathematics, vol. 554. Springer, Berlin (1976)Google Scholar