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  1. Paul Bankston (2011). Base-Free Formulas in the Lattice-Theoretic Study of Compacta. Archive for Mathematical Logic 50 (5-6):531-542.
    The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta (...)
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  2. Paul Bankston (2011). On the First-Order Expressibility of Lattice Properties Related to Unicoherence in Continua. Archive for Mathematical Logic 50 (3-4):503-512.
    Many properties of compacta have “textbook” definitions which are phrased in lattice-theoretic terms that, ostensibly, apply only to the full closed-set lattice of a space. We provide a simple criterion for identifying such definitions that may be paraphrased in terms that apply to all lattice bases of the space, thereby making model-theoretic tools available to study the defined properties. In this note we are primarily interested in properties of continua related to unicoherence; i.e., properties that speak to the existence of (...)
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  3. Paul Bankston (2006). The Chang-Łoś-Suszko Theorem in a Topological Setting. Archive for Mathematical Logic 45 (1):97-112.
    The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.
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  4. Paul Bankston (1999). A Hierarchy of Maps Between Compacta. Journal of Symbolic Logic 64 (4):1628-1644.
    Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev ≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev ≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank (...)
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  5. Paul Bankston (1991). Corrigendum to "Taxonomies of Model-Theoretically Defined Topological Properties". Journal of Symbolic Logic 56 (2):425-426.
    An error has been found in the cited paper; namely, Theorem 3.1 is false.
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  6. Paul Bankston (1990). Taxonomies of Model-Theoretically Defined Topological Properties. Journal of Symbolic Logic 55 (2):589-603.
    A topological classification scheme consists of two ingredients: (1) an abstract class K of topological spaces; and (2) a "taxonomy", i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class. K is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed (...)
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  7. Paul Bankston & Wim Ruitenburg (1990). Notions of Relative Ubiquity for Invariant Sets of Relational Structures. Journal of Symbolic Logic 55 (3):948-986.
    Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, (...)
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  8. Paul Bankston (1987). Reduced Coproducts of Compact Hausdorff Spaces. Journal of Symbolic Logic 52 (2):404-424.
    By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with (...)
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  9. Paul Bankston (1984). Expressive Power in First Order Topology. Journal of Symbolic Logic 49 (2):478-487.
    A first order representation (f.o.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions "one f.o.r. is at least as expressive as another relative to a class of spaces" and "one class of spaces is definable in another relative to an f.o.r.", and prove some general statements. Following this we compare some well-known classes of spaces and (...)
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