Finite Abelian algebras are dualizable

TL;DR

This paper proves that finite Abelian algebras are dualizable by showing that their compatible relations are entailed by relations of bounded arity, extending previous results on modules and solving an open problem.

Contribution

It establishes dualizability of finite Abelian algebras using a relation-arity entailment approach, improving upon prior results for modules and finite algebras.

Findings

Finite Abelian algebras are dualizable.

Relations of bounded arity entail all compatible relations.

Dualizing alter-ego can have relations of arity ≤ 1+α^3.

Abstract

A finite algebra $\bA=\alg{A;\cF}$ is \emph{dualizable} if there exists a discrete topological relational structure $\BA=\alg{A;\cG;\cT}$, compatible with $\cF$, such that the canonical evaluation map $e\_{\bB}\colon \bB\to \Hom( \Hom(\bB,\bA),\BA)$ is an isomorphism for every $\bB$ in the quasivariety generated by $\bA$. Here, $e\_{\bB}$ is defined by $e\_{\bB}(x)(f)=f(x)$ for all $x\in B$ and all $f\in \Hom(\bB,\bA)$. We prove that, given a finite congruence-modular Abelian algebra $\bA$, the set of all relations compatible with $\bA$, up to a certain arity, \emph{entails} the whole set of all relations compatible with $\bA$. By using a classical compactness result, we infer that $\bA$ is dualizable. Moreover we can choose a dualizing alter-ego with only relations of arity $\le 1+\alpha^3$, where $\alpha$ is the largest exponent of a prime in the prime decomposition of $\card{A}$. This improves Kearnes and Szendrei result that modules are dualizable, and Bentz and Mayr's result that finite modules with constants are dualizable. This also solves a problem stated by Bentz and Mayr in 2013.