Supernilpotence prevents dualizability

Bentz, Wolfram; Mayr, Peter

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.

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

• Journal Article Type: Article
• Acceptance Date: Aug 8, 2013
• Online Publication Date: Sep 30, 2013
• Publication Date: 2014-02
• Deposit Date: Jun 29, 2018
• Publicly Available Date: Jul 25, 2018
• Journal: Journal of the Australian Mathematical Society
• Print ISSN: 1446-7887
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 96
• Issue: 01
• Pages: 1-24
• DOI: https://doi.org/10.1017/S1446788713000517
• Public URL: https://hull-repository.worktribe.com/output/901232
• Publisher URL: https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/supernilpotence-prevents-dualizability/19216A9906A81512881DD0E2E67BBEE8
• Contract Date: Jun 29, 2018