Supernilpotent Taylor algebras are nilpotent

Moorhead, Andrew

doi:10.1090/tran/8251

Supernilpotent Taylor algebras are nilpotent
HTML articles powered by AMS MathViewer

by Andrew Moorhead PDF

Trans. Amer. Math. Soc. 374 (2021), 1229-1276 Request permission

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.

References

Erhard Aichinger, Bounding the free spectrum of nilpotent algebras of prime power order, Israel J. Math. 230 (2019), no. 2, 919–947. MR 3940441, DOI 10.1007/s11856-019-1846-x
Erhard Aichinger and Nebojša Mudrinski, Some applications of higher commutators in Mal’cev algebras, Algebra Universalis 63 (2010), no. 4, 367–403. MR 2734303, DOI 10.1007/s00012-010-0084-1
Andrei A. Bulatov, A dichotomy theorem for nonuniform CSPs, 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2017, IEEE Computer Soc., Los Alamitos, CA, 2017, pp. 319–330. MR 3734240, DOI 10.1109/FOCS.2017.37
Andrei Bulatov, On the number of finite Mal′tsev algebras, Contributions to general algebra, 13 (Velké Karlovice, 1999/Dresden, 2000) Heyn, Klagenfurt, 2001, pp. 41–54. MR 1854568
Ralph Freese and Ralph McKenzie, Commutator theory for congruence modular varieties, London Mathematical Society Lecture Note Series, vol. 125, Cambridge University Press, Cambridge, 1987. MR 909290
O. C. García and W. Taylor, The lattice of interpretability types of varieties, Mem. Amer. Math. Soc. 50 (1984), no. 305, v+125. MR 749524, DOI 10.1090/memo/0305
H. Peter Gumm, Geometrical methods in congruence modular algebras, Mem. Amer. Math. Soc. 45 (1983), no. 286, viii+79. MR 714648, DOI 10.1090/memo/0286
Joachim Hagemann and Christian Herrmann, A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. (Basel) 32 (1979), no. 3, 234–245. MR 541622, DOI 10.1007/BF01238496
Christian Herrmann, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 119–125. MR 534504
PawełM. Idziak and Jacek Krzaczkowski, Satisfiability in multi-valued circuits, LICS ’18—33rd Annual ACM/IEEE Symposium on Logic in Computer Science, ACM, New York, 2018, pp. [9 pp.]. MR 3883762, DOI 10.1145/3209108.3209173
K. A. Kearnes, Congruence modular varieties with small free spectra, Algebra Universalis 42 (1999), no. 3, 165–181. MR 1736712, DOI 10.1007/s000120050132
Keith A. Kearnes and Emil W. Kiss, The shape of congruence lattices, Mem. Amer. Math. Soc. 222 (2013), no. 1046, viii+169. MR 3076179, DOI 10.1090/S0065-9266-2012-00667-8
Keith A. Kearnes and Agnes Szendrei, Is supernilpotence super nilpotence?, manuscript in preparation.
Keith A. Kearnes and Ágnes Szendrei, The relationship between two commutators, Internat. J. Algebra Comput. 8 (1998), no. 4, 497–531. MR 1663558, DOI 10.1142/S0218196798000247
Michael Kompatscher, The equation solvability problem over supernilpotent algebras with Mal’cev term, Internat. J. Algebra Comput. 28 (2018), no. 6, 1005–1015. MR 3855005, DOI 10.1142/S0218196718500443
A. I. Mal′cev, On the general theory of algebraic systems, Amer. Math. Soc. Transl. (2) 27 (1963), 125–142. MR 0151416
Matthew Moore and Andrew Moorhead, Supernilpotence need not imply nilpotence, J. Algebra 535 (2019), 225–250. MR 3979090, DOI 10.1016/j.jalgebra.2019.07.002
Andrew Moorhead, Higher commutator theory for congruence modular varieties, J. Algebra 513 (2018), 133–158. MR 3849882, DOI 10.1016/j.jalgebra.2018.07.026
Andrew Moorhead, Some notes on the ternary modular commutator, ArXiv e-prints (2018), Available at arXiv:1808.01407.
Jakub Opršal, A relational description of higher commutators in Mal’cev varieties, Algebra Universalis 76 (2016), no. 3, 367–383. MR 3556818, DOI 10.1007/s00012-016-0391-2
Jonathan D. H. Smith, Mal′cev varieties, Lecture Notes in Mathematics, Vol. 554, Springer-Verlag, Berlin-New York, 1976. MR 0432511
Walter Taylor, Varieties obeying homotopy laws, Canadian J. Math. 29 (1977), no. 3, 498–527. MR 434928, DOI 10.4153/CJM-1977-054-9
Alexander Wires, On supernilpotent algebras, Algebra Universalis 80 (2019), no. 1, Paper No. 1, 37. MR 3892989, DOI 10.1007/s00012-018-0574-0
Dmitriy Zhuk, A proof of CSP dichotomy conjecture, 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2017, IEEE Computer Soc., Los Alamitos, CA, 2017, pp. 331–342. MR 3734241, DOI 10.1109/FOCS.2017.38