Finite Abelian algebras are fully dualizable

TL;DR

This paper proves that all finite Abelian algebras in congruence-permutable varieties admit a full duality, with explicit bounds and structural properties, advancing the understanding of their dualizability.

Contribution

It establishes full dualizability for finite Abelian algebras in congruence-permutable varieties and provides explicit bounds on duality structures.

Findings

Finite Abelian algebras admit a full duality.

The duality can be induced by a finite type dualizing structure.

The enriched partial hom-clone is finitely generated.

Abstract

We show that every finite Abelian algebra A from congruence-permutable varieties admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure of finite type. We give an explicit bound on the arities of the partial and total operations appearing in the dualizing structure. In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.