Search results for 'Algebra' (try it on Scholar)

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  1. Aldo Ursini, Paolo Aglianò, Roberto Magari & International Conference on Logic and Algebra (1996). Logic and Algebra.
     
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  2. Paolo Mancosu & Richard Zach (2015). Heinrich Behmann's 1921 Lecture on the Decision Problem and the Algebra of Logic. Bulletin of Symbolic Logic 21 (2):164-187.
    Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented (...)
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  3.  40
    I. C. Baianu, R. Brown, G. Georgescu & J. F. Glazebrook (2006). Complex Non-Linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks. [REVIEW] Axiomathes 16 (1-2):65-122.
    A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...)
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  4.  22
    Tamar Lando (2012). Completeness of S4 for the Lebesgue Measure Algebra. Journal of Philosophical Logic 41 (2):287-316.
    We prove completeness of the propositional modal logic S 4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and . Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure (...), , and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in with an open representative. We prove completeness of the modal logic S 4 for the algebra . A corollary to the main result is that non-theorems of S 4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in. (shrink)
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  5.  15
    Camilo Argoty (2013). The Model Theory of Modules of a C*-Algebra. Archive for Mathematical Logic 52 (5-6):525-541.
    We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show that there is an (...)
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  6.  79
    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 properties of (...)
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  7.  42
    K. Muralidhar (2014). Complex Vector Formalism of Harmonic Oscillator in Geometric Algebra: Particle Mass, Spin and Dynamics in Complex Vector Space. Foundations of Physics 44 (3):266-295.
    Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown (...)
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  8.  19
    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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  9.  6
    Gido Scharfenberger-Fabian (2011). Souslin Algebra Embeddings. Archive for Mathematical Logic 50 (1-2):75-113.
    A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part (...)
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  10.  33
    Sho Tanaka (2009). Kinematical Reduction of Spatial Degrees of Freedom and Holographic Relation in Yang's Quantized Space-Time Algebra. Foundations of Physics 39 (5):510-518.
    We try to find a possible origin of the holographic principle in the Lorentz-covariant Yang’s quantized space-time algebra (YSTA). YSTA, which is intrinsically equipped with short- and long-scale parameters, λ and R, gives a finite number of spatial degrees of freedom for any bounded spatial region, providing a basis for divergence-free quantum field theory. Furthermore, it gives a definite kinematical reduction of spatial degrees of freedom, compared with the ordinary lattice space. On account of the latter fact, we (...)
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  11.  10
    Daniele Mundici (2006). A Characterization of the Free N-Generated MV-Algebra. Archive for Mathematical Logic 45 (2):239-247.
    An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function (...)
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  12.  60
    Mirja Hartimo (2012). Husserl and the Algebra of Logic: Husserl's 1896 Lectures. [REVIEW] Axiomathes 22 (1):121-133.
    In his 1896 lecture course on logic–reportedly a blueprint for the Prolegomena to Pure Logic –Husserl develops an explicit account of logic as an independent and purely theoretical discipline. According to Husserl, such a theory is needed for the foundations of logic (in a more general sense) to avoid psychologism in logic. The present paper shows that Husserl’s conception of logic (in a strict sense) belongs to the algebra of logic tradition. Husserl’s conception is modeled after arithmetic, and (...)
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  13.  13
    Albrecht Heeffer (2008). The Emergence of Symbolic Algebra as a Shift in Predominant Models. Foundations of Science 13 (2):149--161.
    Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This (...)
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  14.  18
    Jeffrey A. Oaks (2007). Medieval Arabic Algebra as an Artificial Language. Journal of Indian Philosophy 35 (5-6):543-575.
    Medieval Arabic algebra is a good example of an artificial language.Yet despite its abstract, formal structure, its utility was restricted to problem solving. Geometry was the branch of mathematics used for expressing theories. While algebra was an art concerned with finding specific unknown numbers, geometry dealtwith generalmagnitudes.Algebra did possess the generosity needed to raise it to a more theoretical level—in the ninth century Abū Kāmil reinterpreted the algebraic unknown “thing” to prove a general result. (...)
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  15.  17
    P. Watson & A. J. Bracken (2014). Quantum Phase Space From Schwinger's Measurement Algebra. Foundations of Physics 44 (7):762-780.
    Schwinger’s algebra of microscopic measurement, with the associated complex field of transformation functions, is shown to provide the foundation for a discrete quantum phase space of known type, equipped with a Wigner function and a star product. Discrete position and momentum variables label points in the phase space, each taking \(N\) distinct values, where \(N\) is any chosen prime number. Because of the direct physical interpretation of the measurement symbols, the phase space structure is thereby related to definite (...)
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  16.  4
    M. Alizadeh & M. Ardeshir (2004). On the Linear Lindenbaum Algebra of Basic Propositional Logic. Mathematical Logic Quarterly 50 (1):65.
    We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra.
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  17.  14
    Neil Thapen & Michael Soltys (2005). Weak Theories of Linear Algebra. Archive for Mathematical Logic 44 (2):195-208.
    We investigate the theories of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment in which we can interpret a weak theory V 1 of bounded arithmetic and carry out polynomial time reasoning about matrices - for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, proves (...)
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  18.  2
    L. P. Belluce, Antonio Di Nola & Salvatore Sessa (1994). The Prime Spectrum of an MV‐Algebra. Mathematical Logic Quarterly 40 (3):331-346.
    In this paper we show that the prime ideal space of an MV-algebra is the disjoint union of prime ideal spaces of suitable local MV-algebras. Some special classes of algebras are defined and their spaces are investigated. The space of minimal prime ideals is studied as well.
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  19.  5
    Ivo D.[Uuml ]Ntsch, Gunther Schmidt & Michael Winter (2001). A Necessary Relation Algebra for Mereotopology. 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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  20.  10
    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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  21.  14
    David J. Foulis & Sylvia Pulmannová (2013). Type-Decomposition of a Synaptic Algebra. Foundations of Physics 43 (8):948-968.
    A synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. In this article we extend to synaptic algebras the type-I/II/III decomposition of von Neumann algebras, AW∗-algebras, and JW-algebras.
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  22.  33
    Yingxu Wang (2003). Using Process Algebra to Describe Human and Software Behaviors. Brain and Mind 4 (2):199-213.
    Although there are various ways to express actions and behaviors in natural languages, it is found in cognitive informatics that human and system behaviors may be classified into three basic categories: to be , to have , and to do . All mathematical means and forms, in general, are an abstract description of these three categories of system behaviors and their common rules. Taking this view, mathematical logic may be perceived as the abstract means for describing to be, set theory (...)
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  23.  3
    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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  24.  19
    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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  25.  9
    Imants Barušs & Robert Woodrow (2013). A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category Can Always Be Replaced by Forcing Over a Complete Heyting Algebra. [REVIEW] Logica Universalis 7 (3):323-334.
    It is assumed that a Kripke–Joyal semantics \({\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}\) has been defined for a first-order language \({\mathcal{L}}\) . To transform \({\mathbb{C}}\) into a Heyting algebra \({\overline{\mathbb{C}}}\) on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of \({\mathbb{C}}\) . A pretopology \({\overline{{\rm Cov}}}\) is defined on \({\overline{\mathbb{C}}}\) using the pretopology on \({\mathbb{C}}\) . A sheaf \({\overline{{\it F}}}\) is made up of (...)
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  26.  3
    Brendan Larvor (2005). Proof in C17 Algebra. Philosophia Scientae:43-59.
    By the middle of the seventeenth century we that find that algebra is able to offer proofs in its own right. That is, by that time algebraic argument had achieved the status of proof. How did this transformation come about?
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  27.  7
    Jie Fang, Lei-Bo Wang & Ting Yang (2014). The Lattice of Kernel Ideals of a Balanced Pseudocomplemented Ockham Algebra. Studia Logica 102 (1):29-39.
    In this note we shall show that if L is a balanced pseudocomplemented Ockham algebra then the set ${\fancyscript{I}_{k}(L)}$ of kernel ideals of L is a Heyting lattice that is isomorphic to the lattice of congruences on B(L) where ${B(L) = \{x^* | x \in L\}}$ . In particular, we show that ${\fancyscript{I}_{k}(L)}$ is boolean if and only if B(L) is finite, if and only if every kernel ideal of L is principal.
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  28.  14
    Dirk Leinders, Maarten Marx, Jerzy Tyszkiewicz & Jan Van den Bussche (2005). The Semijoin Algebra and the Guarded Fragment. Journal of Logic, Language and Information 14 (3):331-343.
    In the 1970s Codd introduced the relational algebra, with operators selection, projection, union, difference and product, and showed that it is equivalent to first-order logic. In this paper, we show that if we replace in Codd’s relational algebra the product operator by the “semijoin” operator, then the resulting “semijoin algebra” is equivalent to the guarded fragment of first-order logic. We also define a fixed point extension of the semijoin algebra that corresponds to μGF.
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  29.  10
    Todor D. Todorov & Hans Vernaeve (2008). Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis. Logic and Analysis 1 (3-4):205-234.
    We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a (...)
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  30.  3
    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 the (...)
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  31.  8
    Ivo D.[Uuml ]Ntsch, Gunther Schmidt & Michael Winter (2001). A Necessary Relation Algebra for Mereotopology. 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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  32.  6
    Alex Citkin (2012). Not Every Splitting Heyting or Interior Algebra is Finitely Presentable. Studia Logica 100 (1-2):115-135.
    We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.
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  33.  1
    Gurgen Asatryan (2008). On Models of Exponentiation. Identities in the HSI-Algebra of Posets. Mathematical Logic Quarterly 54 (3):280-287.
    We prove that Wilkie's identity holds in those natural HSI-algebras where each element has finite decomposition into components.Further, we construct a bunch of HSI-algebras that satisfy all the identities of the set of positive integers ℕ. Then, based on the constructed algebras, we prove that the identities of ℕ hold in the HSI-algebra of finite posets when the value of each variable is a poset having an isolated point.
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  34. 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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  35.  5
    Abraham Robinson (1963). Introduction to Model Theory and to the Metamathematics of Algebra. North-Holland.
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  36.  3
    Yunfu Shen (1999). Elimination of Algorithmic Quantifiers for Ordered Differential Algebra. Archive for Mathematical Logic 38 (3):139-144.
    In [2], Singer proved that the theory of ordered differential fields has a model completion, i.e, the theory of closed ordered differential fields, CODF. As a result, CODF admits elimination of quantifiers. In this paper we give an algorithm to eliminate the quantifiers of CODF-formulas.
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  37.  8
    Murdoch J. Gabbay (2012). Finite and Infinite Support in Nominal Algebra and Logic: Nominal Completeness Theorems for Free. Journal of Symbolic Logic 77 (3):828-852.
    By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction. The results are of interest in their own right, (...)
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  38.  8
    David J. Foulis & Sylvia Pulmannová (2010). Type-Decomposition of an Effect Algebra. Foundations of Physics 40 (9-10):1543-1565.
    Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras.We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a (...)
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  39.  3
    Cheryl C. Graesser & Norman H. Anderson (1974). Cognitive Algebra of the Equation: Gift Size = Generosity = Income. Journal of Experimental Psychology 103 (4):692.
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  40.  3
    Kenneth R. Koedinger, Martha W. Alibali & Mitchell J. Nathan (2008). Trade‐Offs Between Grounded and Abstract Representations: Evidence From Algebra Problem Solving. Cognitive Science 32 (2):366-397.
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  41. John N. Crossley (ed.) (1975). Algebra and Logic: Papers From the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia. Springer-Verlag.
     
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  42. John Gregg (1998). Ones and Zeros Understanding Boolean Algebra, Digital Circuits, and the Logic of Sets. Monograph Collection (Matt - Pseudo).
     
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  43.  1
    Theodore Hailperin (1976). Boole's Logic and Probability a Critical Exposition From the Standpoint of Contemporary Algebra, Logic, and Probability Theory. Monograph Collection (Matt - Pseudo).
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  44.  5
    A. F. Lavrik (ed.) (1979). Twelve Papers in Logic and Algebra. American Mathematical Society.
    (2) VoL ll3, l979 PROPERTIES OF SOME SUBSYSTEMS OF CLASSICAL AND INTUITIONISTIC PROPOSITIONAL CALCULI* MI SEMENENKO Preface In both classical and ...
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  45. Allan Whitcombe, Alan Boxer, Maureen Donaldson & David Wright (1993). Logic and Algebra.
     
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  46. Thomas Mormann, The Contact Algebra of the Euclidean Plane has Infinitely Many Elements.
    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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  47. Daniel Sutherland (2006). Kant on Arithmetic, Algebra, and the Theory of Proportions. Journal of the History of Philosophy 44 (4):533-558.
    Daniel Sutherland - Kant on Arithmetic, Algebra, and the Theory of Proportions - Journal of the History of Philosophy 44:4 Journal of the History of Philosophy 44.4 533-558 Muse Search Journals This Journal Contents Kant on Arithmetic, Algebra, and the Theory of Proportions Daniel Sutherland Kant's philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account of his (...)
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  48. Harvey Friedman, A Consistency Proof for Elementary Algebra and Geometry.
    We give a consistency proof within a weak fragment of arithmetic of elementary algebra and geometry. For this purpose, we use EFA (exponential function arithmetic), and various first order theories of algebraically closed fields and real closed fields.
     
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  49. Alexander Berenstein (2004). Dividing in the Algebra of Compact Operators. Journal of Symbolic Logic 69 (3):817-829.
    We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.
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  50. 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, in (...)
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