Search results for 'Boolean algebra' (try it on Scholar)

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  1.  35
    Pablo F. Castro & Piotr Kulicki (forthcoming). Deontic Logics Based on Boolean Algebra. 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 (...)
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  2.  4
    J. Donald Monk (2001). The Spectrum of Partitions of a Boolean Algebra. Archive for Mathematical Logic 40 (4):243-254.
    The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a (...)
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  3.  0
    Matatyahu Rubin & Sabine Koppelberg (2001). A Superatomic Boolean Algebra with Few Automorphisms. Archive for Mathematical Logic 40 (2):125-129.
    Assuming GCH, we prove that for every successor cardinal μ > ω1, there is a superatomic Boolean algebra B such that |B| = 2μ and |Aut B| = μ. Under ◊ω1, the same holds for μ = ω1. This answers Monk's Question 80 in [Mo].
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  4.  4
    James Cummings & Saharon Shelah (1995). A Model in Which Every Boolean Algebra has Many Subalgebras. Journal of Symbolic Logic 60 (3):992-1004.
    We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2 |A| = 2 |B| . This implies in particular that B has 2 |B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper (...)
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  5. Aldo Ursini, Paolo Aglianò, Roberto Magari & International Conference on Logic and Algebra (1996). Logic and Algebra.
     
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  6. John Gregg (1998). Ones and Zeros Understanding Boolean Algebra, Digital Circuits, and the Logic of Sets. Monograph Collection (Matt - Pseudo).
     
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  7.  8
    Robert Trypuz & Piotr Kulicki (2010). A Systematics of Deontic Action Logics Based on Boolean Algebra. Logic and Logical Philosophy 18 (3-4):253-270.
    Within the scope of interest of deontic logic, systems in which names of actions are arguments of deontic operators (deontic action logic) have attracted less interest than purely propositional systems. However, in our opinion, they are even more interesting from both theoretical and practical point of view. The fundament for contemporary research was established by K. Segerberg, who introduced his systems of basic deontic logic of urn model actions in early 1980s. Nowadays such logics are considered mainly within propositional dynamic (...)
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  8.  15
    Lei-Bo Wang (2010). Congruences on a Balanced Pseudocomplemented Ockham Algebra Whose Quotient Algebras Are Boolean. Studia Logica 96 (3):421-431.
    In this note we shall describe the lattice of the congruences on a balanced Ockham algebra with the pseudocomplementation whose quotient algebras are boolean. This is an extension of the result obtained by Rodrigues and Silva who gave a description of the lattice of congruences on an Ockham algebra whose quotient algebras are boolean.
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  9.  2
    Ivo Düntsch & Sanjiang Li (2013). On the Homogeneous Countable Boolean Contact Algebra. 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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  10.  49
    E. W. Madison (1983). The Existence of Countable Totally Nonconstructive Extensions of the Countable Atomless Boolean Algebra. Journal of Symbolic Logic 48 (1):167-170.
    Our results concern the existence of a countable extension U of the countable atomless Boolean algebra B such that U is a "nonconstructive" extension of B. It is known that for any fixed admissible indexing φ of B there is a countable nonconstructive extension U of B (relative to φ). The main theorem here shows that there exists an extension U of B such that for any admissible indexing φ of B, U is nonconstructive (relative to φ). Thus, (...)
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  11. Uwe Meixner (1998). Negative Theology, Coincidentia Oppositorum, and Boolean Algebra. Logical Analysis and History of Philosophy 1:75-89.
    In Plato's Parmenides we find on the one hand that the One is denied every property , and on the other hand that the One is attributed every property . In the course of the history of Platonism , these assertions - probably meant by Plato as ontological statements of an entirely formal nature - were repeatedly made the starting points of metaphysical speculations. In the Mystical Theology of the Pseudo-Dionysius they became principles of Christian mysticism and negative theology. I (...)
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  12.  23
    V. A. Bocharov (1986). Boolean Algebra and Syllogism. Synthese 66 (1):35 - 54.
    This article contains the proof of equivalence boolean algebra and syllogistics arc2. The system arc2 is obtained as a superstructure above the propositional calculus. Subjects and predicates of syllogistic functors a, E, J, O may be complex terms, Which are formed using operations of intersection, Union and complement. In contrast to negative sentences the interpretation of affirmative sentences suggests non-Empty terms. To prove the corresponding theorem we demonstrate that boolean algebra is included into syllogistics arc2 and (...)
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  13.  4
    J. Donald Monk (2004). The Spectrum of Maximal Independent Subsets of a Boolean Algebra. Annals of Pure and Applied Logic 126 (1-3):335-348.
    Recall that a subset X of a Boolean algebra A is independent if for any two finite disjoint subsets F , G of X we have ∏ x∈F x ∏ y∈G −y≠0. The independence of a BA A , denoted by Ind, is the supremum of cardinalities of its independent subsets. We can also consider the maximal independent subsets. The smallest size of an infinite maximal independent subset is the cardinal invariant i , well known in the case (...)
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  14.  20
    Wlesław Dziobiak (1982). Concerning Axiomatizability of the Quasivariety Generated by a Finite Heyting or Topological Boolean Algebra. Studia Logica 41 (4):415 - 428.
    In classes of algebras such as lattices, groups, and rings, there are finite algebras which individually generate quasivarieties which are not finitely axiomatizable (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattice join of two finitely axiomatizable quasivarieties, each generated by a finite Heyting or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, we solve (...)
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  15.  9
    Daniel D. Merrill (2005). Augustus De Morgan's Boolean Algebra. History and Philosophy of Logic 26 (2):75-91.
    De Morgan's Formal Logic, which was published on virtually the same day in 1847 as Boole's The Mathematical Analysis of Logic, contains a logic of complex terms (LCT) which has been sadly neglected. It is surprising to find that LCT contains almost a full theory of Boolean algebra. This paper will: (1) provide some background to LCT; (2) outline its main features; (3) point out some gaps in it; (4) compare it with Boole's algebra; (5) show that (...)
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  16.  9
    E. W. Madison & B. Zimmermann-Huisgen (1986). Combinatorial and Recursive Aspects of the Automorphism Group of the Countable Atomless Boolean Algebra. Journal of Symbolic Logic 51 (2):292-301.
    Given an admissible indexing φ of the countable atomless Boolean algebra B, an automorphism F of B is said to be recursively presented (relative to φ) if there exists a recursive function $p \in \operatorname{Sym}(\omega)$ such that F ⚬ φ = φ ⚬ p. Our key result on recursiveness: Both the subset of $\operatorname{Aut}(\mathscr{B})$ consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all "totally nonrecursive" automorphisms, are uncountable. (...)
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  17.  3
    Claude Sureson (2007). Rumely Domains with Atomic Constructible Boolean Algebra. An Effective Viewpoint. Notre Dame Journal of Formal Logic 48 (3):399-423.
    The archetypal Rumely domain is the ring \widetildeZ of algebraic integers. Its constructible Boolean algebra is atomless. We study here the opposite situation: Rumely domains whose constructible Boolean algebra is atomic. Recursive models (which are rings of algebraic numbers) are proposed; effective model-completeness and decidability of the corresponding theory are proved.
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  18.  0
    Hirokazu Nishimura (1994). Boolean Valued and Stone Algebra Valued Measure Theories. 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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  19.  4
    Don H. Faust (1982). The Boolean Algebra of Formulas of First-Order Logic. Annals of Mathematical Logic 23 (1):27-53.
    The algebraic recursive structure of countable languages of classical first-order logic with equality is analysed. all languages of finite undecidable similarity type are shown to be algebraically and recursively equivalent in the following sense: their boolean algebras of formulas are, after trivial factors involving the one element models of the languages have been excepted, recursively isomorphic by a map which preserves the degree of recursiveness of their models.
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  20.  5
    Marcel Erné (2009). Finiteness Conditions and Distributive Laws for Boolean Algebras. 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 converse (...)
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  21.  3
    Roman Wencel (2012). Imaginaries in Boolean Algebras. Mathematical Logic Quarterly 58 (3):217-235.
    Given an infinite Boolean algebra B, we find a natural class of equation image-definable equivalence relations equation image such that every imaginary element from Beq is interdefinable with an element from a sort determined by some equivalence relation from equation image. It follows that B together with the family of sorts determined by equation image admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries (...)
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  22.  3
    Juan Carlos Martínez & Lajos Soukup (2011). Superatomic Boolean Algebras Constructed From Strongly Unbounded Functions. 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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  23.  3
    V. Yu Shavrukov (2010). Effectively Inseparable Boolean Algebras in Lattices of Sentences. Archive for Mathematical Logic 49 (1):69-89.
    We show the non-arithmeticity of 1st order theories of lattices of Σ n sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on (...)
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  24.  10
    Thomas Jech & Saharon Shelah (1996). On Countably Closed Complete Boolean Algebras. Journal of Symbolic Logic 61 (4):1380-1386.
    It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.
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  25.  1
    Paul Howard (2011). The Finiteness of Compact Boolean Algebras. 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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  26.  0
    J. Donald Monk (1996). Minimum‐Sized Infinite Partitions of Boolean Algebras. Mathematical Logic Quarterly 42 (1):537-550.
    For any Boolean Algebra A, let cmm be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, (...)
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  27.  0
    Markus Huberich (1996). A Note on Boolean Algebras with Few Partitions Modulo Some Filter. Mathematical Logic Quarterly 42 (1):172-174.
    We show that for every uncountable regular κ and every κ-complete Boolean algebra B of density ≤ κ there is a filter F ⊆ B such that the number of partitions of length < modulo κF is ≤2<κ. We apply this to Boolean algebras of the form P/I, where I is a κ-complete κ-dense ideal on X.
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  28.  0
    M. S. Kurilic (2004). Unsupported Boolean Algebras and Forcing. Mathematical Logic Quarterly 50 (6):594.
    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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  29.  0
    Makoto Takahashi & Yasuo Yoshinobu (2003). Σ‐Short Boolean Algebras. 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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  30.  0
    Carlo Toffalori & S. Leonesi (2001). Weakly o-Minimal Expansions of Boolean Algebras. Mathematical Logic Quarterly 47 (2):223-238.
    We propose a definition of weak o-minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o-minimal non o-minimal examples in this setting.
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  31.  1
    Anastasis Kamburelis (1989). Iterations of Boolean Algebras with Measure. Archive for Mathematical Logic 29 (1):21-28.
    We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.
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  32.  1
    Regina Aragón (1995). Some Boolean Algebras with Finitely Many Distinguished Ideals I. 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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  33.  0
    Rod Downey (1993). Every Recursive Boolean Algebra is Isomorphic to One with Incomplete Atoms. Annals of Pure and Applied Logic 60 (3):193-206.
    The theorem of the title is proven, solving an old question of Remmel. The method of proof uses an algebraic technique of Remmel-Vaught combined with a complex tree of strategies argument where the true path is needed to figure out the final isomorphism.
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  34.  2
    Cecylis Rauszer & Bogdan Sabalski (1974). Representation Theorem for Distributive Pseudo-Boolean Algebra. Bulletin of the Section of Logic 3 (3/4):17-21.
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  35.  8
    Luis M. Laita (1980). Boolean Algebra and its Extra-Logical Sources: The Testimony of Mary Everest Boole. History and Philosophy of Logic 1 (1-2):37-60.
    Mary Everest, Boole's wife, claimed after the death of her husband that his logic had a psychological, pedagogical, and religious origin and aim rather than the mathematico-logical ones assigned to it by critics and scientists. It is the purpose of this paper to examine the validity of such a claim. The first section consists of an exposition of the claim without discussing its truthfulness; the discussion is left for the sections 2?4, in which some arguments provided by the examination of (...)
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  36.  1
    Alonzo Church (1937). Review: Albert Whiteman, Postulates for Boolean Algebra in Terms of Ternary Rejection. [REVIEW] Journal of Symbolic Logic 2 (2):91-91.
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  37.  30
    Robert E. Clay (1974). Relation of Leśniewski's Mereology to Boolean Algebra. Journal of Symbolic Logic 39 (4):638-648.
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  38. Martin Gardner (1958/1968). Logic Machines, Diagrams and Boolean Algebra. New York, Dover Publications.
     
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  39.  2
    R. C. Lyndon (1951). Review: Arthur H. Copeland, Implicative Boolean Algebra. [REVIEW] Journal of Symbolic Logic 16 (2):151-152.
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  40.  1
    Thomas H. Mott (1962). Review: Franz E. Hohn, Applied Boolean Algebra. An Elementary Introduction. [REVIEW] Journal of Symbolic Logic 27 (1):104-106.
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  41.  1
    Bogus law Wolniewicz (1981). The Boolean Algebra of Objectives. Bulletin of the Section of Logic 10 (1):17-22.
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  42.  1
    H. P. K. (1968). Boolean Algebra. Review of Metaphysics 21 (4):751-751.
  43.  4
    Mystery Of Measurability (2006). Is a Set B with Boolean Operations a∨ B (Join), a∧ B (Meet) and− a (Complement), Partial Ordering a≤ B Defined by a∧ B= a and the Smallest and Greatest Element, 0 and 1. By Stone's Representation Theorem, Every Boolean Algebra is Isomorphic to an Algebra of Subsets of Some Nonempty Set S, Under Operations a∪ B, a∩ B, S− a, Ordered by Inclusion, with 0=∅. [REVIEW] Bulletin of Symbolic Logic 12 (2).
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  44.  5
    Raymond J. Nelson (1955). Review: D. E. Muller, Application of Boolean Algebra to Switching Circuit Design and to Error Detection. [REVIEW] Journal of Symbolic Logic 20 (2):195-195.
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  45.  2
    Thomas H. Mott (1962). Review: J. Eldon Whitesitt, Boolean Algebra and Its Applications. [REVIEW] Journal of Symbolic Logic 27 (1):103-104.
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  46.  2
    Robert E. Clay (1984). Relation of Leśniewski's Mereology to Boolean Algebra. In Jan T. J. Srzednicki, V. F. Rickey & J. Czelakowski (eds.), Journal of Symbolic Logic. Distributors for the United States and Canada, Kluwer Boston 241--252.
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  47.  4
    Boguslaw Wolniewicz (1981). The Boolean Algebra of Objectives. Bulletin of the Section of Logic 10 (1):17-22.
    This is the fth and last installment in series dealing with the Wittgen- steinian notion of a situation . All proofs and most lemmas have been omitted. They are contained in a comprehensive paper on the ontol- ogy of situations to be submitted to Studia Logica.
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  48.  12
    Archie Blake (1938). Corrections to Canonical Expressions in Boolean Algebra. Journal of Symbolic Logic 3 (3):112-113.
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  49.  1
    K. E. Aubert (1959). Review: Arthur H. Copeland, Frank Harary, The Extension of an Arbitrary Boolean Algebra to an Implicative Boolean Algebra. [REVIEW] Journal of Symbolic Logic 24 (3):254-254.
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  50.  1
    G. Gratzer (1972). Review: J. C. Abbott, Semi-Boolean Algebra. [REVIEW] Journal of Symbolic Logic 37 (1):191-191.
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