2015/03/09 by Pierre Gillibert, Gillibert, Pierre
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1503.02651
arxiv created 2015/03/09 · openalex publication_date 2015/03/09 · arxiv updated 2015/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A finite algebra \bA=\algA;\cF is dualizable if there exists a discrete topological relational structure \BA=\algA;\cG;\cT, compatible with \cF, such that the canonical evaluation map e_\bB\colon \bB→ \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∈ B and all f∈ \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, 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 ≤ 1+α3, where α is the largest exponent of a prime in the prime decomposition of \cardA. 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.