Finite Abelian algebras and dualizable
09/05/2014 Sexta-feira, 9 de Maio de 2014, 14h30m, Sala A2-25
Pierre Gillibert (CAUL)
Institute for Interdisciplinary Research - University of Lisbon
Consider an algebra M=(A,F), that is A is a set and F is a set of operations over A. We say that M is dualizable if there exists a discrete topological relational structure N=(A,G,T), compatible with F, such that the canonical evaluation map e : B -> Hom(Hom(B,M), N) is an isomorphism for every B in the quasivariety generated by M. Here, e is defined by e(x)(f) = f(x) for all x in B and all f in Hom(B,M).
We prove that, given a finite Abelian algebra M=(A,F), the set of all relations compatible with F, up to a certain arity, entails the whole set of all relations compatible with F. By using a classical compactness result, we infer that A is dualizable. This result solves a problem stated by Mayr in 2013. This also extends Bentz and Mayr’s result that finite modules with constants are dualizable.