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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. 

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