Results for 'Boolean contact algebras'

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  1.  38
    On the Homogeneous Countable Boolean Contact Algebra.Ivo Düntsch & Sanjiang Li - 2013 - Logic and Logical Philosophy 22 (2):213-251.
    In a recent paper, we have shown that the class of Boolean contact algebras (BCAs) has the hereditary property, the joint embedding property and the amalgamation property. By Fraïssé’s theorem, this shows that there is a unique countable homogeneous BCA. This paper investigates this algebra and the relation algebra generated by its contact relation. We first show that the algebra can be partitioned into four sets {0}, {1}, K, and L, which are the only orbits of (...)
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  2.  16
    Grzegorczyk Points and Filters in Boolean Contact Algebras.Rafał Gruszczyński & Andrzej Pietruszczak - 2023 - Review of Symbolic Logic 16 (2):509-528.
    The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last (...)
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  3. The Contact Algebra of the Euclidean Plane has Infinitely Many Elements.Thomas Mormann - manuscript
    Abstract. Let REL(O*E) be the relation algebra of binary relations defined on the Boolean algebra O*E of regular open regions of the Euclidean plane E. The aim of this paper is to prove that the canonical contact relation C of O*E generates a subalgebra REL(O*E, C) of REL(O*E) that has infinitely many elements. More precisely, REL(O*,C) contains an infinite family {SPPn, n ≥ 1} of relations generated by the relation SPP (Separable Proper Part). This relation can be used (...)
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  4.  29
    Extended Contact Algebras and Internal Connectedness.Tatyana Ivanova - 2020 - Studia Logica 108 (2):239-254.
    The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C, called contact. Standard models of contact algebras are topological and are the contact algebras of regular closed sets in a given topological space. In such a contact algebra we add the predicate of internal connectedness with the following meaning—a regular closed set is internally (...)
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  5. Review: A. G. Lunc, The Application of Boolean Matrix Algebra to the Analysis and Synthesis of Relay-Contact Networks. [REVIEW]Zdzislaw Pawlak - 1956 - Journal of Symbolic Logic 21 (1):104-104.
  6.  23
    Relational Representation Theorems for Extended Contact Algebras.Philippe Balbiani & Tatyana Ivanova - 2020 - Studia Logica 109 (4):701-723.
    In topological spaces, the relation of extended contact is a ternary relation that holds between regular closed subsets A, B and D if the intersection of A and B is included in D. The algebraic counterpart of this mereotopological relation is the notion of extended contact algebra which is a Boolean algebra extended with a ternary relation. In this paper, we are interested in the relational representation theory for extended contact algebras. In this respect, we (...)
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  7.  10
    Contact Join-semilattices.Tatyana Ivanova - 2022 - Studia Logica 110 (5):1219-1241.
    Contact algebra is one of the main tools in region-based theory of space. In it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. Thus we obtain structures, called contact join-semilattices and structures, called distributive contact join-semilattices. We obtain a set-theoretical representation theorem for CJS and a relational representation theorem for DCJS. As corollaries we get also topological representation theorems. We prove that the universal (...)
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  8.  19
    From Contact Relations to Modal Operators, and Back.Rafał Gruszczyński & Paula Menchón - 2023 - Studia Logica 111 (5):717-748.
    One of the standard axioms for Boolean contact algebras says that if a region __x__ is in contact with the join of __y__ and __z__, then __x__ is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if __x__ is in contact with the supremum of some family __S__ of regions, then there is a __y__ in __S__ that is (...)
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  9.  24
    Logics for extended distributive contact lattices.T. Ivanova - 2018 - Journal of Applied Non-Classical Logics 28 (1):140-162.
    The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. There are some problems related to the motivation of the operation of Boolean complementation. Because of this operation is dropped and the language of distributive lattices is extended by considering as non-definable primitives the relations of contact, nontangential inclusion and dual contact. It is (...)
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  10.  23
    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 (...)
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  11.  10
    Contact semilattices.Paolo Lipparini - forthcoming - Logic Journal of the IGPL.
    We devise exact conditions under which a join semilattice with a weak contact relation can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact relation. A similar characterization is proved with respect to Boolean algebras and distributive lattices with weak contact, not necessarily additive, nor overlap.
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  12.  33
    A Necessary Relation Algebra for Mereotopology.Michael Winter, Gunther Schmidt & Ivo DÜntsch - 2001 - Studia Logica 69 (3):381-409.
    The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T0 topological space with an additional "contact relation" C defined by xCy ? x n ? Ø.
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  13.  44
    A mereotopology based on sequent algebras.Dimiter Vakarelov - 2017 - Journal of Applied Non-Classical Logics 27 (3-4):342-364.
    Mereotopology is an extension of mereology with some relations of topological nature like contact. An algebraic counterpart of mereotopology is the notion of contact algebra which is a Boolean algebra whose elements are considered to denote spatial regions, extended with a binary relation of contact between regions. Although the language of contact algebra is quite expressive to define many useful mereological relations and mereotopological relations, there are, however, some interesting mereotopological relations which are not definable (...)
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  14. A necessary relation algebra for mereotopology.Ivo DÜntsch, Gunther Schmidt & Michael Winter - 2001 - Studia Logica 69 (3):381 - 409.
    The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T 0 topological space with an additional "contact relation" C defined by xCy x ØA (possibly) more general class of models is provided by the Region Connection Calculus (RCC) of Randell et al. We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any (...)
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  15.  13
    Boolean connection algebras: A new approach to the Region-Connection Calculus.J. G. Stell - 2000 - Artificial Intelligence 122 (1-2):111-136.
  16.  4
    Boolean-Like Algebras of Finite Dimension: From Boolean Products to Semiring Products.Antonio Bucciarelli, Antonio Ledda, Francesco Paoli & Antonino Salibra - 2024 - In Jacek Malinowski & Rafał Palczewski (eds.), Janusz Czelakowski on Logical Consequence. Springer Verlag. pp. 377-400.
    We continue the investigation, initiated in Salibra et al. (Found Sci, 2020), of Boolean-like algebras of dimension n (nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s), algebras having n constants e1,⋯,en\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf e_1,\dots,\mathsf e_n$$\end{document}, and an (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of (...)
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  17.  31
    Boolean sentence algebras: Isomorphism constructions.William P. Hanf & Dale Myers - 1983 - Journal of Symbolic Logic 48 (2):329-338.
    Associated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional (...)
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  18.  11
    Mathematical Methods in Region-Based Theories of Space: The Case of Whitehead Points.Rafał Gruszczyński - 2024 - Bulletin of the Section of Logic 53 (1):63-104.
    Regions-based theories of space aim—among others—to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed, one should show that they are points of a geometrical or a topological space constructed in a specific way. This paper intends to show how the development of mathematical tools allows showing that Whitehead’s method of extensive abstraction provides a construction of objects that are fundamental (...)
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  19.  12
    Boolean subtractive algebras.Thomas M. Hearne & Carl G. Wagner - 1974 - Notre Dame Journal of Formal Logic 15 (2):317-324.
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  20.  20
    Normal and complete Boolean ambiguity algebras and MV-pairs.H. de la Vega - 2012 - Logic Journal of the IGPL 20 (6):1133-1152.
  21.  11
    Outlines of a Boolean Tensor Algebra with Applications to the Lower Functional Calculus.Hakan Tornebohm - 1960 - Journal of Symbolic Logic 25 (4):367-368.
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  22.  41
    Outlines of a Boolean tensor algebra with applications to the lower functional calculus.Håkan Törnebohm - 1958 - Theoria 24 (1):39-47.
  23.  25
    The Boolean Sentence Algebra of the Theory of Linear Ordering is Atomic with Respect to Logics with a Malitz Quantifier.Hans-Joachim Goltz - 1985 - Mathematical Logic Quarterly 31 (9-12):131-162.
  24.  34
    The Boolean Sentence Algebra of the Theory of Linear Ordering is Atomic with Respect to Logics with a Malitz Quantifier.Hans-Joachim Goltz - 1985 - Mathematical Logic Quarterly 31 (9-12):131-162.
  25.  29
    Foundations of Boolean Valued Algebraic Geometry.Hirokazu Nishimura - 1991 - Mathematical Logic Quarterly 37 (26‐30):421-438.
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  26.  36
    Foundations of Boolean Valued Algebraic Geometry.Hirokazu Nishimura - 1991 - Mathematical Logic Quarterly 37 (26-30):421-438.
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  27.  12
    Chains in Boolean Semigroup Algebras.Lutz Heindorf - 1991 - Mathematical Logic Quarterly 37 (5‐6):93-96.
  28.  26
    Chains in Boolean Semigroup Algebras.Lutz Heindorf - 1991 - Mathematical Logic Quarterly 37 (5-6):93-96.
  29.  23
    Admissibility of Π2-Inference Rules: interpolation, model completion, and contact algebras.Nick Bezhanishvili, Luca Carai, Silvio Ghilardi & Lucia Landi - 2023 - Annals of Pure and Applied Logic 174 (1):103169.
  30.  62
    Logics based on partial Boolean sigma-algebras.Janusz Czelakowski - 1975 - Studia Logica 34:69.
  31.  45
    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 up (...)
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  32.  36
    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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  33. Deontic Logics based on Boolean Algebra.Pablo F. Castro & Piotr Kulicki - forthcoming - In Robert Trypuz (ed.), Krister Segerberg on Logic of Actions. Springer.
    Deontic logic is devoted to the study of logical properties of normative predicates such as permission, obligation and prohibition. Since it is usual to apply these predicates to actions, many deontic logicians have proposed formalisms where actions and action combinators are present. Some standard action combinators are action conjunction, choice between actions and not doing a given action. These combinators resemble boolean operators, and therefore the theory of boolean algebra offers a well-known athematical framework to study the properties (...)
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  34.  18
    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 (...)
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  35.  37
    Boolean deductive systems of BL-algebras.Esko Turunen - 2001 - Archive for Mathematical Logic 40 (6):467-473.
    BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hájek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems (...)
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  36.  9
    Decidability of topological quasi-Boolean algebras.Yiheng Wang, Zhe Lin & Minghui Ma - 2024 - Journal of Applied Non-Classical Logics 34 (2):269-293.
    A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S5 and prove that the cut elimination holds for (...)
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  37.  31
    Boolean products of R0-algebras.Xiangnan Zhou & Qingguo Li - 2010 - Mathematical Logic Quarterly 56 (3):289-298.
    In this paper, the Boolean representation of R0-algebras are investigated. In particular, we show that directly indecomposable R0-algebras are equivalent to local R0-algebras and any nontrivial R0-algebra is representable as a weak Boolean product of local R0-algebras.
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  38.  34
    Overview of Finite Propositional Boolean Algebras I.Branden Fitelson - unknown
    of monadic or relational predicate calculus (Fa, Gb, Rab, Hcd, etc.). • The Boolean Algebra BL set-up by such a language will be such that: – BL will have 2 n states (corresponding to the state descriptions of L) – BL will contain 2 2n propositions, in total. ∗ This is because each proposition p in BL is equivalent to a disjunction of state descriptions. Thus, each subset of the set of..
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  39.  26
    Review: F. E. J. Linton, Injective Boolean $Sigma$-Algebras[REVIEW]Adil Yaqub - 1972 - Journal of Symbolic Logic 37 (2):400-400.
  40.  20
    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 (...)
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  41.  35
    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.  15
    Free Boolean algebras and nowhere dense ultrafilters.Aleksander Błaszczyk - 2004 - Annals of Pure and Applied Logic 126 (1-3):287-292.
    An analogue of Mathias forcing is studied in connection of free Boolean algebras and nowhere dense ultrafilters. Some applications to rigid Boolean algebras are given.
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  43.  19
    Boolean Skeletons of MV-algebras and ℓ-groups.Roberto Cignoli - 2011 - Studia Logica 98 (1-2):141-147.
    Let Γ be Mundici’s functor from the category $${\mathcal{LG}}$$ whose objects are the lattice-ordered abelian groups ( ℓ -groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category $${\mathcal{MV}}$$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ℓ -group G , the Boolean skeleton of the MV-algebra Γ ( G , u ) is isomorphic to the Boolean algebra of factor (...)
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  44. Some Boolean Algebras with Finitely Many Distinguished Ideals I.Regina Arag N. - 1995 - Mathematical Logic Quarterly 41 (4):485-504.
     
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  45.  13
    Törnebohm Håkan. Outlines of a Boolean tensor algebra with applications to the lower functional calculus. Theoria , vol. 24 , pp. 39–47. [REVIEW]H. Arnold Schmidt - 1960 - Journal of Symbolic Logic 25 (4):367-368.
  46.  52
    Partial Boolean algebras in a broader sense.Janusz Czelakowski - 1979 - Studia Logica 38 (1):1 - 16.
    The article deals with compatible families of Boolean algebras. We define the notion of a partial Boolean algebra in a broader sense (PBA(bs)) and then we show that there is a mutual correspondence between PBA(bs) and compatible families of Boolean algebras (Theorem (1.8)). We examine in detail the interdependence between PBA(bs) and the following classes: partial Boolean algebras in the sense of Kochen and Specker (§ 2), ortholattices (§ 3, § 5), and orthomodular (...)
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  47.  38
    A boolean transfer principle from L*‐Algebras to AL*‐Algebras.Hirokazu Nishimura - 1993 - Mathematical Logic Quarterly 39 (1):241-250.
    Just as Kaplansky [4] has introduced the notion of an AW*-module as a generalization of a complex Hilbert space, we introduce the notion of an AL*-algebra, which is a generalization of that of an L*-algebra invented by Schue [9, 10]. By using Boolean valued methods developed by Ozawa [6–8], Takeuti [11–13] and others, we establish its basic properties including a fundamental structure theorem. This paper should be regarded as a continuation or our previous paper [5], the familiarity with which (...)
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  48.  33
    Boolean Valued and Stone Algebra Valued Measure Theories.Hirokazu Nishimura - 1994 - Mathematical Logic Quarterly 40 (1):69-75.
    In conventional generalization of the main results of classical measure theory to Stone algebra valued measures, the values that measures and functions can take are Booleanized, while the classical notion of a σ-field is retained. The main purpose of this paper is to show by abundace of illustrations that if we agree to Booleanize the notion of a σ-field as well, then all the glorious legacy of classical measure theory is preserved completely.
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  49.  26
    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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  50.  21
    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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