Finite Abelian algebras are dualizable

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 $e_{bB}colon bBto Hom( Hom(bB,bA),BA)$ 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 $xin B$ and all $fin Hom(bB,bA)$. 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 Mayrs result that finite modules with constants are dualizable. This also solves a problem stated by Bentz and Mayr in 2013.