The axiomatizability of topological prevarieties

Abstract

We investigate first-order axiomatic descriptions of naturally occurring classes of Boolean topological structures (these structures can have operations and relations, and carry a compatible compact Hausdorff topology with a basis of clopen sets). Our methods utilize inverse limits and ultraproducts of finite structures. We illustrate the range of possible axiomatizations of these classes with applications of our 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, we find the class of continuously k-colorable Boolean topological graphs is not even first-order axiomatizable.

Boolean topological graphs of semigroups: the lack of first-order axiomatization

“…In particular, it holds when H is a variety of semigroups, monoids, groups, rings or is a variety with definable principal congruences. Now Lemma 2.8 of [6] shows that since H is a variety, the members of H are actually inverse limits of finite members of H , whence H is standard.…”

“…The importance of this notion comes from the fact that if X is pointwise nonseparable with respect to H , then X / ∈ H BC [6,Lemma 3.3]. In the proof of Proposition 4.1 we will use the following fact.…”

“…Let us formulate this as the second specific problem (in the finitely generated case it is Problem 3 in [6]). …”

“…This was done by finding, for each n, a structure X n / ∈ H such that all n-element substructures of X n are in H . The method used in the present article to demonstrate non-standardness is Theorem 4.2 taken from [6], which requires that there exist structures X n having the property just described and, additionally, that these structures can be assembled into an inverse limit satisfying certain conditions. Thus inverse limits satisfying the conditions of Theorem 4.2 witness both the non-standardness and the non-finite axiomatizability of a universal Horn class.…”

“…Every finite structure X when equipped with the discrete topology becomes a topological structure X . Following Clark, Davey, Jackson and Pitkethly [6], we define the Boolean core H BC of a universal Horn class H as the topological prevariety generated by the finite members of H (treated as topological structures). Notably, H BC consists of all profinite structures built, as inverse limits, from finite members of H [6, Corollary 2.4].…”