The axiomatizability of topological prevarieties
type: Item
instance of: scholarly article
title: The axiomatizability of topological prevarieties
zbMATH Open document ID: 1148.08003
published in: Advances in Mathematics
publication date: 2008-07-10T00:00:00Z
review text: The original motivation for this study comes from the theory of natural dualities, which is a tool for studying finitely generated quasivarieties of algebras. A natural duality provides us with a dual equivalence between the quasivariety generated by a finite algebra and a special dual category of compact topological structures. In order to utilize a natural duality fully, it is generally necessary that it be accompanied by an effective description of the objects in the dual category. This dual category is generated by some finite discretely topologized structure \(M\), being the class \(IS_{c}P^{+}(M) \) of all isomorphic copies of topologically closed substructures of non-zero direct powers (with the product topology) of \(M.\) In this paper the authors investigate first-order axiomatic descriptions of naturally occurring classes of Boolean topological structures. These methods utilize inverse limits and ultraproducts of finite structures and illustrate the range of possible axiomatizations of these classes with applications of methods to Boolean topological lattices, graphs, ordered structures, unary algebras and semigroups. For example, whereas the class of all \(k\)-colorable graphs is known to be axiomatizable by universal Horn sentences; the authors find the class of continuously \(k\)-colorable Boolean topological graphs is not even first-order axiomatizable.
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 03C07
Mathematics Subject Classification ID: 03C13
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 54H12
zbMATH DE Number: 5299373
zbMATH Keywords: natural duality
zbMATH Keywords: universal Horn class
zbMATH Keywords: topological quasivariety
zbMATH Keywords: topological prevariety
zbMATH Keywords: profinite
zbMATH Keywords: inverse limit
zbMATH Keywords: lattice
zbMATH Keywords: ordered set
zbMATH Keywords: graph
zbMATH Keywords: unary algebra
zbMATH Keywords: semigroup
author: David M. Clark
author: Brian A. Davey
author: Marcel Jackson
author: Jane G. Pitkethly
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
full work available at URL: j.aim.2008.03.020
OpenAlex ID: W2030229759
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DOI: 10.1016/J.AIM.2008.03.020
description: scientific article; zbMATH DE number 5299373