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Supernilpotence prevents dualizability

Bentz, Wolfram; Mayr, Peter

Authors

Peter Mayr



Abstract

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.

Journal Article Type Article
Publication Date 2014-02
Journal Journal of the Australian Mathematical Society
Print ISSN 1446-7887
Electronic ISSN 1446-8107
Publisher Cambridge University Press (CUP)
Peer Reviewed Peer Reviewed
Volume 96
Issue 01
Pages 1-24
APA6 Citation Bentz, W., & Mayr, P. (2014). Supernilpotence prevents dualizability. Journal of the Australian Mathematical Society, 96(01), 1-24. https://doi.org/10.1017/S1446788713000517
DOI https://doi.org/10.1017/S1446788713000517
Publisher URL https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/supernilpotence-prevents-dualizability/19216A9906A81512881DD0E2E67BBEE8
Copyright Statement This article has been published in a revised form in Journal of the Australian Mathematical Society http://doi.org/10.1017/S1446788713000517. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2013 Australian Mathematical Publishing Association Inc.

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Copyright Statement
This article has been published in a revised form in Journal of the Australian Mathematical Society http://doi.org/10.1017/S1446788713000517. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2013 Australian Mathematical Publishing Association Inc.





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