Finite Abelian algebras are dualizable

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.