The axiomatizability of topological prevarieties
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.
scientific article; zbMATH DE number 5299373
scientific article