Supernilpotence prevents dualizability

TL;DR

This paper investigates the dualizability of nilpotent Mal'cev algebras, demonstrating that supernilpotence prevents dualizability in finite cases, but providing a novel example of a non-abelian nilpotent dualizable algebra.

Contribution

It introduces the first known non-abelian nilpotent dualizable algebra and highlights the role of supernilpotence in dualizability of Mal'cev algebras.

Findings

Finite nilpotent non-abelian Mal'cev algebras with supernilpotent congruences are non-dualizable.

A new example of an infinite-type, non-abelian, nilpotent dualizable algebra is constructed.

Supernilpotence is key in understanding dualizability among Mal'cev algebras.

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.