Abstract
AbstractWe 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.
Highlights
Natural dualities are representations of elements of an algebra as continuous structure preserving maps
While Mal’cev algebras are in general considered to be well understood, the question of their dualizability has so far only been addressed for various classes of classical algebras
We refer the reader to Clark and Davey [4] for background on natural duality
Summary
Natural dualities are representations of elements of an algebra as continuous structure preserving maps (obtained in a certain natural way). While Mal’cev algebras are in general considered to be well understood, the question of their dualizability has so far only been addressed for various classes of classical algebras (see below for examples). The finite nonabelian loops whose multiplication group (the group generated by all left and right translations) is nilpotent are an application of the theorem to a class of algebras that were not previously known to be nondualizable ([16, Proposition 3.2], see [2, Corollary III, page 282]). There we will obtain a more general nondualizability result for nilpotent algebras with a nonabelian, supernilpotent congruence (Theorem 3.1). We will define this notion of supernilpotence (a stronger condition than nilpotence), give equivalent formulations of the conditions of the theorem, and prove several auxiliary results on Mal’cev algebras. We refer the reader to Clark and Davey [4] for background on natural duality
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