STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Pure and Applied Algebra Année : 2016

STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION

Résumé

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognizable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in computer science is intrinsic to profinite completion in mathematics in general. This connection underlies a number of recent results in automata theory including a generalization of Eilenberg-Reiterman theory for regular languages and a new notion of compact recognition applying beyond the setting of regular languages. The purpose of this paper is to give the underlying duality theoretic result in its general form. Further we illustrate the power of the result by providing a few applications in topological algebra and language theory. In particular, we give a simple proof of the fact that any topological algebra quotient of a profinite algebra which is again based on a Boolean space is in fact a profinite algebra and we derive the conditions dual to the ones of the original Eilenberg theorem in a fully modular manner. We cast our results in the setting of extended Priestley duality for distributive lattices with additional operations as some classes of languages of interest in automata fail to be closed under complementation.
Fichier principal
Vignette du fichier
Ge2015.pdf (460.02 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00859717 , version 1 (09-09-2013)
hal-00859717 , version 2 (10-09-2013)
hal-00859717 , version 3 (24-11-2015)
hal-00859717 , version 4 (02-12-2015)

Identifiants

Citer

Mai Gehrke. STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION. Journal of Pure and Applied Algebra, 2016. ⟨hal-00859717v4⟩
913 Consultations
1316 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More