Results for 'Compact Boolean algebra'

959 found
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  1.  26
    The finiteness of compact Boolean algebras.Paul Howard - 2011 - Mathematical Logic Quarterly 57 (1):14-18.
    We show that it consistent with Zermelo-Fraenkel set theory that there is an infinite, compact Boolean algebra.
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  2.  21
    Partition Complete Boolean Algebras and Almost Compact Cardinals.Peter Jipsen & Henry Rose - 1999 - Mathematical Logic Quarterly 45 (2):241-255.
    For an infinite cardinal K a stronger version of K-distributivity for Boolean algebras, called k-partition completeness, is defined and investigated . It is shown that every k-partition complete Boolean algebra is K-weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K-partition complete Boolean algebra is an updirected union of basic K-tree algebras. Using K-partition completeness, the concept of γ-almost compactness is introduced for γ (...)
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  3.  52
    Superatomic Boolean algebras constructed from strongly unbounded functions.Juan Carlos Martínez & Lajos Soukup - 2011 - Mathematical Logic Quarterly 57 (5):456-469.
    Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra equation image such that equation image, equation image for every α < η and equation image. Especially, equation image and equation (...)
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  4.  36
    On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem löwenheim theorems and compactness of related quantifiers.Matatyahu Rubin & Saharon Shelah - 1980 - Journal of Symbolic Logic 45 (2):265-283.
    THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ (...)
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  5.  29
    Property {(hbar)} and cellularity of complete Boolean algebras.Miloš S. Kurilić & Stevo Todorčević - 2009 - Archive for Mathematical Logic 48 (8):705-718.
    A complete Boolean algebra ${\mathbb{B}}$ satisfies property ${(\hbar)}$ iff each sequence x in ${\mathbb{B}}$ has a subsequence y such that the equality lim sup z n = lim sup y n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we (...)
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  6.  43
    On the compactness of some Boolean algebras.Jacek Cichoń - 1984 - Journal of Symbolic Logic 49 (1):63-67.
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  7.  24
    Spectra of Quasi-Boolean Algebras.Yajie Lv & Wenjuan Chen - forthcoming - Logic Journal of the IGPL.
    In the present paper, we introduce the notions of quasi-Boolean algebras as the generalization of Boolean algebras. First we discuss the related properties of quasi-Boolean algebras. Second we define filters of quasi-Boolean algebras and investigate some properties of filters in quasi-Boolean algebras. We also show that there is a one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra. Then we investigate the prime filters and maximal (...)
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  8.  35
    Strictly positive measures on Boolean algebras.Mirna Džamonja & Grzegorz Plebanek - 2008 - Journal of Symbolic Logic 73 (4):1416-1432.
    We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable separable positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the (...)
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  9.  18
    Finiteness conditions and distributive laws for Boolean algebras.Marcel Erné - 2009 - Mathematical Logic Quarterly 55 (6):572-586.
    We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom implies finiteness of Boolean algebras with compact top, whereas the (...)
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  10.  36
    A posteriori convergence in complete Boolean algebras with the sequential topology.Miloš S. Kurilić & Aleksandar Pavlović - 2007 - Annals of Pure and Applied Logic 148 (1-3):49-62.
    A sequence x=xn:nω of elements of a complete Boolean algebra converges to a priori if lim infx=lim supx=b. The sequential topology τs on is the maximal topology on such that x→b implies x→τsb, where →τs denotes the convergence in the space — the a posteriori convergence. These two forms of convergence, as well as the properties of the sequential topology related to forcing, are investigated. So, the a posteriori convergence is described in terms of killing of tall ideals (...)
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  11.  82
    Alexander Abian. On the solvability of infinite systems of Boolean polynomial equations. Colloquium mathematicum, vol. 21 , pp. 27–30. - Alexander Abian. Generalized completeness theorem and solvability of systems of Boolean polynomial equations. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 16 , pp. 263–264. - Paul D. Bacsich. Injectivity in model theory. Colloquium mathematicum, vol. 25 , pp. 165–176. - S. Bulman-Fleming. On equationally compact semilattices. Algebra universalis , vol. 2 no. 2 , pp. 146–151. - G. Grätzer and H. Lakser. Equationally compact semilattices. Colloquium mathematicum, vol. 20 , pp. 27–30. - David K. Haley. On compact commutative Noetherian rings. Mathematische Annalen, vol. 189 , pp. 272–274. - Ralph McKenzie. ℵ1-incompactness of Z. Colloquium mathematicum, vol. 23 , pp. 199–202. - Jan Mycielski. Some compactifications of general algebras. Colloquium mathematicum, vol. 13 no. 1 , pp. 1–9. See Errata on page 281 of next paper. - Jan. [REVIEW]Walter Taylor - 1975 - Journal of Symbolic Logic 40 (1):88-92.
  12. An algebraic treatment of the Barwise compactness theory.Isidore Fleischer & Philip Scott - 1991 - Studia Logica 50 (2):217 - 223.
    A theorem on the extendability of certain subsets of a Boolean algebra to ultrafilters which preserve countably many infinite meets (generalizing Rasiowa-Sikorski) is used to pinpoint the mechanism of the Barwise proof in a way which bypasses the set theoretical elaborations.
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  13.  64
    Reduced coproducts of compact hausdorff spaces.Paul Bankston - 1987 - 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 (...)
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  14.  90
    Algebraic and topological semantics for inquisitive logic via choice-free duality.Nick Bezhanishvili, Gianluca Grilletti & Wesley H. Holliday - 2019 - In Rosalie Iemhoff, Michael Moortgat & Ruy de Queiroz, Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science, Vol. 11541. Springer. pp. 35-52.
    We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show (...)
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  15.  89
    (1 other version)Boolean universes above Boolean models.Friedrich Wehrung - 1993 - Journal of Symbolic Logic 58 (4):1219-1250.
    We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are "boundedly algebraically compact" in the language $(+,-,\cdot,\wedge,\vee,\leq)$ , and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an (...)
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  16.  30
    Chain conditions of products, and weakly compact cardinals.Assaf Rinot - 2014 - Bulletin of Symbolic Logic 20 (3):293-314,.
    The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an (...)
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  17. Álgebras booleanas, órdenes parciales y axioma de elección.Franklin Galindo - 2017 - Divulgaciones Matematicas 18 ( 1):34-54.
    El objetivo de este artículo es presentar una demostración de un teorema clásico sobre álgebras booleanas y ordenes parciales de relevancia actual en teoría de conjuntos, como por ejemplo, para aplicaciones del método de construcción de modelos llamado “forcing” (con álgebras booleanas completas o con órdenes parciales). El teorema que se prueba es el siguiente: “Todo orden parcial se puede extender a una única álgebra booleana completa (salvo isomorfismo)”. Donde extender significa “sumergir densamente”. Tal demostración se realiza utilizando cortaduras de (...)
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  18.  41
    A variety of algebras closely related to subordination algebras.Sergio Celani & Ramon Jansana - 2022 - Journal of Applied Non-Classical Logics 32 (2):200-238.
    We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et (...)
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  19.  75
    Duality for lattice-ordered algebras and for normal algebraizable logics.Chrysafis Hartonas - 1997 - Studia Logica 58 (3):403-450.
    Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and (...)
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  20. Choice-free stone duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any (...)
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  21.  34
    On completeness of the quotient algebras {cal P}(kappa)/I.Yasuo Kanai - 2000 - Archive for Mathematical Logic 39 (2):75-87.
    In this paper, the following are proved:Theorem A. The quotient algebra ${\cal P} (\kappa )/I$ is complete if and only if the only non-trivial I -closed ideals extending I are of the form $I\lceil A$ for some $A\in I^+$ .Theorem B. If $\kappa$ is a stationary cardinal, then the quotient algebra ${\cal P} (\kappa )/ NS_\kappa$ is not complete.Corollary. (1) If $\kappa$ is a weak compact cardinal, then the quotient algebra ${\cal P} (\kappa )/NS_\kappa$ is not (...)
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  22. Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s 'Completeness Paper' (1930).Vasil Penchev - 2023 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 15 (1):1-61.
    The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...)
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  23.  21
    Modal Operators on Rings of Continuous Functions.Guram Bezhanishvili, Luca Carai & Patrick J. Morandi - 2022 - Journal of Symbolic Logic 87 (4):1322-1348.
    It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with (...)
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  24.  32
    Topological Proofs of Some Rasiowa-Sikorski Lemmas.Robert Goldblatt - 2012 - Studia Logica 100 (1-2):175-191.
    We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.
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  25.  55
    Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem.R. E. Hodel - 2005 - Archive for Mathematical Logic 44 (4):459-472.
    We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)+ and (rTT)++; each is equivalent to (rTT) in ZF. The variation (rTT)++ applies rather naturally to (...)
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  26.  28
    On constructions with 2-cardinals.Piotr Koszmider - 2017 - Archive for Mathematical Logic 56 (7-8):849-876.
    We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman’s neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. The (...)
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  27. On the Gettier Problem for Topological Logic of Knowledge and Belief.Thomas Mormann - manuscript
    Abstract. Gettier’s famous examples intended to show that knowledge cannot always be equated with justified true belief. The Gettier problem can also be considered as a problem for topological epistemic logic: If knowledge and justified belief are conceived as topological operators K and B on topological spaces (to be considered as universes of possible worlds), one may ask whether it happens that there is a proposition A such that KA ≠ A & BA or not. If this is the case, (...)
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  28.  26
    The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters.Eric J. Hall, Kyriakos Keremedis & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (4-5):258-267.
    Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a (...)
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  29.  86
    Analytic ideals.Sławomir Solecki - 1996 - Bulletin of Symbolic Logic 2 (3):339-348.
    §1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters. There is also a substantial interest in nicely definable ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally in (...)
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  30.  48
    Sublocales in Formal Topology.Steven Vickers - 2007 - Journal of Symbolic Logic 72 (2):463 - 482.
    The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated (...)
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  31.  51
    Boolean Algebras in Visser Algebras.Majid Alizadeh, Mohammad Ardeshir & Wim Ruitenburg - 2016 - Notre Dame Journal of Formal Logic 57 (1):141-150.
    We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras. This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.
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  32.  37
    Satisfiability on hypergraphs.Adam Kolany - 1993 - Studia Logica 52 (3):393-404.
    In [4] R.Cowen considers a generalization of the resolution rule for hypergraphs and introduces a notion of satisfiability of families of sets of vertices via 2-colorings piercing elements of such families. He shows, for finite hypergraphs with no one-element edges that if the empty set is a consequence ofA by the resolution rule, thenA is not satisfiable. Alas the converse is true for a restricted class of hypergraphs only, and need not to be true in the general case. In this (...)
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  33.  43
    The mathematics of logic: a guide to completeness theorems and their applications.Richard Kaye - 2007 - New York: Cambridge University Press.
    This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to (...)
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  34.  34
    The Logic ILP for Intuitionistic Reasoning About Probability.Angelina Ilić-Stepić, Zoran Ognjanović & Aleksandar Perović - 2024 - Studia Logica 112 (5):987-1017.
    We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form H,μ\langle H, \mu \rangle that needs not be a probability space. More precisely, though _H_ needs not be a Boolean algebra, the corresponding monotone function (we call it measure) μ:H[0,1]Q\mu : H \longrightarrow [0,1]_{\mathbb {Q}} satisfies the following condition: if α\alpha , β\beta , \(\alpha \wedge (...)
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  35.  85
    Valueless Measures on Pointless Spaces.Tamar Lando - 2022 - Journal of Philosophical Logic 52 (1):1-52.
    On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary relation (‘qualitative probability’) that is tied (...)
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  36.  64
    Boolean Algebras, Tarski Invariants, and Index Sets.Barbara F. Csima, Antonio Montalbán & Richard A. Shore - 2006 - Notre Dame Journal of Formal Logic 47 (1):1-23.
    Tarski defined a way of assigning to each Boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from ℕ, such that two Boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a Boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra (...)
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  37.  9
    From real-life to very strong axioms. Classification problems in Descriptive Set Theory and regularity properties in Generalized Descriptive Set Theory.Martina Iannella - 2024 - Bulletin of Symbolic Logic 30 (2):285-286.
    This thesis is divided into three parts, the first and second ones focused on combinatorics and classification problems on discrete and geometrical objects in the context of descriptive set theory, and the third one on generalized descriptive set theory at singular cardinals of countable cofinality.Descriptive Set Theory (briefly: DST) is the study of definable subsets of Polish spaces, i.e., separable completely metrizable spaces. One of the major branches of DST is Borel reducibility, successfully used in the last 30 years to (...)
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  38.  18
    The Baire Closure and its Logic.G. Bezhanishvili & D. Fernández-Duque - 2024 - Journal of Symbolic Logic 89 (1):27-49.
    The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and (...)
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  39.  35
    The space of minimal structures.Oleg Belegradek - 2014 - Mathematical Logic Quarterly 60 (1-2):40-53.
    For a signature L with at least one constant symbol, an L‐structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L‐structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier‐free L‐sentences, and therefore one can define a Stone topology on. This topology on generalizes the topology of the space of n‐marked groups. We introduce a natural ultrametric on, and show that the Stone topology (...)
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  40.  33
    Cardinal sequences.István Juhász & William Weiss - 2006 - Annals of Pure and Applied Logic 144 (1-3):96-106.
    In this article we characterize all those sequences of cardinals of length ω1 which are cardinal sequences of some compact scattered space . This extends the similar results from [R. La Grange, Concerning the cardinal sequence of a Boolean algebra, Algebra Universalis, 7 307–313] for such sequences of countable length. For ordinals between ω1 and ω2 we can only give a sufficient condition for a sequence of that length to be a cardinal sequence of a (...) scattered space. This condition is, arguably, the most one can expect in ZFC. In any case, ours is a significant extension of the sufficient conditions given in [J.C. Martinez, A consistency result on thin-tall superatomic Boolean algebras, Proc. Amer. Math. Soc. 115 473–477] and [J. Bagaria, Locally generic Boolean algebras and cardinal sequences, Algebra Universalis 47 283–302]. (shrink)
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  41.  47
    Boolean Algebras and Distributive Lattices Treated Constructively.John L. Bell - 1999 - Mathematical Logic Quarterly 45 (1):135-143.
    Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.
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  42. Boolean Algebra as the Basis of Mathematical Logic.John-Michael Kuczynski - 2016 - Madison, WI, USA: Philosophypedia.
    The theorems of the propositional calculus and the predicate calculus are stated, and the analogous principles of Boolean Algebra are identified. Also, the primary principles of modal logic are stated, and a procedure is described for identifying their Boolean analogues.
     
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  43.  51
    σ-short Boolean algebras.Makoto Takahashi & Yasuo Yoshinobu - 2003 - Mathematical Logic Quarterly 49 (6):543-549.
    We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach-Mazur Boolean game. A σ-short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ-short Boolean algebras and study properties of σ-short Boolean algebras.
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  44.  20
    Unsupported Boolean algebras and forcing.Miloš S. Kurilić - 2004 - Mathematical Logic Quarterly 50 (6):594-602.
    If κ is an infinite cardinal, a complete Boolean algebra B is called κ-supported if for each sequence 〈bβ : β αbβ = equation imagemath imageequation imageβ∈Abβ holds. Combinatorial and forcing equivalents of this property are given and compared with the other forcing related properties of Boolean algebras . The set of regular cardinals κ for which B is not κ-supported is investigated.
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  45.  35
    (1 other version)Boolean algebras in ast.Klaus Schumacher - 1992 - Mathematical Logic Quarterly 38 (1):373-382.
    In this paper we investigate Boolean algebras and their subalgebras in Alternative Set Theory . We show that any two countable atomless Boolean algebras are isomorphic and we give an example of such a Boolean algebra. One other main result is, that there is an infinite Boolean algebra freely generated by a set. At the end of the paper we show that the sentence “There is no non-trivial free group which is a set” is (...)
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  46.  26
    Some Boolean algebras with finitely many distinguished ideals II.Regina Aragón - 2003 - Mathematical Logic Quarterly 49 (3):260.
    We describe the countably saturated models and prime models of the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the (...)
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  47.  28
    Some Boolean Algebras with Finitely Many Distinguished Ideals I.Regina Aragón - 1995 - Mathematical Logic Quarterly 41 (4):485-504.
    We consider the theory Thprin of Boolean algebras with a principal ideal, the theory Thmax of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Thsa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Thprin and Thsa. If T is a theory in a first order language and (...)
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  48.  63
    Superatomic Boolean algebras constructed from morasses.Peter Koepke & Juan Carlos Martínez - 1995 - Journal of Symbolic Logic 60 (3):940-951.
    By using the notion of a simplified (κ,1)-morass, we construct κ-thin-tall, κ-thin-thick and, in a forcing extension, κ-very thin-thick superatomic Boolean algebras for every infinite regular cardinal κ.
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  49.  28
    Boolean algebras arising from information systems.Ivo Düntsch & Ewa Orłowska - 2004 - Annals of Pure and Applied Logic 127 (1-3):77-98.
    Following the theory of Boolean algebras with modal operators , in this paper we investigate Boolean algebras with sufficiency operators and mixed operators . We present results concerning representability, generation by finite members, first order axiomatisability, possession of a discriminator term etc. We generalise the classes BAO, SUA, and MIA to classes of algebras with the families of relative operators. We present examples of the discussed classes of algebras that arise in connection with reasoning with incomplete information.
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  50.  59
    Boolean Algebras, Stone Spaces, and the Iterated Turing Jump.Carl G. Jockusch & Robert I. Soare - 1994 - Journal of Symbolic Logic 59 (4):1121 - 1138.
    We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω (...)
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