Supernilpotence prevents dualizability

Abstract

We address the question of the dualizability of nilpotent Mal'cev algebras, showing that nilpotent finite Mal'cev algebras with a non-abelian supernilpotent congruence are inherently non-dualizable. In particular, finite nilpotent non-abelian Mal'cev algebras of finite type are non-dualizable 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 non-abelian, but dualizable. To our knowledge this is the first construction of a non-abelian nilpotent dualizable algebra. It has the curious property that all its non-abelian finitary reducts with group operation are non-dualizable. 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.