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SUPERNILPOTENCE PREVENTS DUALIZABILITY

Part of: Model theory Algebraic structures Varieties Other classes of algebra

Published online by Cambridge University Press:  30 September 2013

WOLFRAM BENTZ  and
PETER MAYR
WOLFRAM BENTZ*
Affiliation:
Centro de Álgebra, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal email [email protected]
PETER MAYR
Affiliation:
Centro de Álgebra, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal email [email protected] Institute for Algebra, Johannes Kepler University Linz, Altenberger Straße 69, 4040 Linz, Austria email [email protected]
*
✉[email protected]
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Abstract

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We address the question of the dualizability of nilpotent Mal’cev algebras, showing that nilpotent finite Mal’cev algebras with a nonabelian supernilpotent congruence are inherently nondualizable. In particular, finite nilpotent nonabelian Mal’cev algebras of finite type are nondualizable if they are direct products of algebras of prime power order. We show that these results cannot be generalized to nilpotent algebras by giving an example of a group expansion of infinite type that is nilpotent and nonabelian, but dualizable. To our knowledge this is the first construction of a nonabelian nilpotent dualizable algebra. It has the curious property that all its nonabelian finitary reducts with group operation are nondualizable. We were able to prove dualizability by utilizing a new clone theoretic approach developed by Davey, Pitkethly, and Willard. Our results suggest that supernilpotence plays an important role in characterizing dualizability among Mal’cev algebras.

Keywords

natural dualityMal’cev algebranilpotencepartial clones

MSC classification

Secondary: 03C05: Equational classes, universal algebra 08A05: Structure theory 08B05: Equational logic, Mal′cev (Mal′tsev) conditions 08C15: Quasivarieties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 1 , February 2014 , pp. 1 - 24
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Aichinger, E. and Mudrinski, N., ‘Some applications of higher commutators in Mal’cev algebras’, Algebra Universalis 63 (4) (2010), 367–403.Google Scholar
Bruck, R. H., ‘Contributions to the theory of loops’, Trans. Amer. Math. Soc. 60 (2) (1946), 245–354.Google Scholar
Bulatov, A., ‘On the number of finite Mal’cev algebras’, in: Proc. Dresden Conference 2000 (AAA 60) and the Summer School 1999, Contr. Gen. Alg., 13 (2001), 41–54.Google Scholar
Clark, D. M. and Davey, B. A., Natural Dualities for the Working Algebraist, Cambridge Studies in Advanced Mathematics, 57 (Cambridge University Press, Cambridge, 1998).Google Scholar
Clark, D. M., Davey, B. A. and Pitkethly, J. G., ‘The complexity of dualisability: three-element unary algebras’, Int. J. Algebra Comput. 13 (3) (2003), 361–391.CrossRefGoogle Scholar
Clark, D. M., Idziak, P. M., Sabourin, L. R., Szabó, Cs. and Willard, Ross, ‘Natural dualities for quasivarieties generated by a finite commutative ring’, Algebra Universalis 46 (1–2) (2001), 285–320; the Viktor Aleksandrovich Gorbunov memorial issue.Google Scholar
Davey, B. A., Heindorf, L. and McKenzie, R., ‘Near unanimity: an obstacle to general duality theory’, Algebra Universalis 33 (1995), 428–439.Google Scholar
Davey, B. A., Pitkethly, J. G. and Willard, R., ‘The lattice of alter egos’, Int. J. Algebra Comput. 22 (1) (2012), 36.Google Scholar
Freese, R. and McKenzie, R. N., Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series, 125 (Cambridge University Press, Cambridge, 1987).Google Scholar
Kearnes, K. A., ‘Congruence modular varieties with small free spectra’, Algebra Universalis 42 (3) (1999), 165–181.Google Scholar
Mayr, P., ‘Mal’cev algebras with supernilpotent centralizers’, Algebra Universalis 65 (2) (2011), 193–211.Google Scholar
McKenzie, R. N., McNulty, G. F. and Taylor, W. F., Algebras, Lattices, Varieties, Vol. I (Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987).Google Scholar
Nickodemus, M. H., ‘Natural dualities for finite groups with abelian Sylow subgroups’. ProQuest LLC, Ann Arbor, MI, 2007, PhD Thesis, University of Colorado at Boulder.Google Scholar
Quackenbush, R. and Szabó, Cs., ‘Nilpotent groups are not dualizable’, J. Aust. Math. Soc. 72 (2) (2002), 173–179.Google Scholar
Szabó, Cs., ‘Finite nilpotent rings are not dualizable’, Algebra Universalis 42 (4) (1999), 293–298.Google Scholar
Vesanen, A., ‘On p-groups as loop groups’, Arch. Math. (Basel) 61 (1993), 1–6.Google Scholar
Willard, R., ‘Four unsolved problems in congruence permutable varieties’, Talk at the Conference on Order, Algebra, and Logics, Nashville, 2007.Google Scholar
Willard, R., ‘New tools for proofing dualizability’, in: Dualities, Interpretability and Ordered Structures (eds. Vaz de Carvalho, J. and Ferreirim, I.) (Centro de Algebra da Universidade de Lisboa, 1999), 69–74.Google Scholar
Zadori, L., ‘Natural Duality via a finite set of relations’, Bull. Aust. Math. Soc. 51 (1995), 469–478.CrossRefGoogle Scholar