Algebra

Edited by Nemi Boris Pelgrom (Ludwig Maximilians Universität, München)
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  1. Algunos tópicos de Lógica matemática y los Fundamentos de la matemática.Franklin Galindo - manuscript
    En este trabajo matemático-filosófico se estudian cuatro tópicos de la Lógica matemática: El método de construcción de modelos llamado Ultraproductos, la Propiedad de Interpolación de Craig, las Álgebras booleanas y los Órdenes parciales separativos. El objetivo principal del mismo es analizar la importancia que tienen dichos tópicos para el estudio de los fundamentos de la matemática, desde el punto de vista del platonismo matemático. Para cumplir con tal objetivo se trabajará en el ámbito de la Matemática, de la Metamatemática y (...)
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  2. The construction of transfinite equivalence algorithms.Han Geurdes - manuscript
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  3. Principles and Philosophy of Linear Algebra: A Gentle Introduction.Paul Mayer - manuscript
    Linear Algebra is an extremely important field that extends everyday concepts about geometry and algebra into higher spaces. This text serves as a gentle motivating introduction to the principles (and philosophy) behind linear algebra. This is aimed at undergraduate students taking a linear algebra class - in particular engineering students who are expected to understand and use linear algebra to build and design things, however it may also prove helpful for philosophy majors and anyone else interested in the ideas behind (...)
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  4. The English Algebra of Logic in the 19th Century.Roman Murawski - unknown - Poznan Studies in the Philosophy of the Sciences and the Humanities 98:245-269.
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  5. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).
    Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)
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  6. The relevance logic of Boolean groups.Yale Weiss - 2023 - Logic Journal of the IGPL 31 (1):96-114.
    In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where negation is included (...)
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  7. Introduction to Neutrosophic Restricted SuperHyperGraphs and Neutrosophic Restricted SuperHyperTrees and several of their properties.Masoud Ghods, Zahra Rostami & Florentin Smarandache - 2022 - Neutrosophic Sets and Systems 50 (1):480-487.
    In this article, we first provide a modified definition of SuperHyperGraphs (SHG) and we call it Restricted SuperHyperGraphs (R-SHG). We then generalize the R-SHG to the neutrosophic graphs and then define the corresponding trees. In the following, we examine the Helly property for subtrees of SuperHyperGraphs.
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  8. Material mathematics : British algebra as algorithmic mathematics.Kevin Lambert - 2022 - In Morgan G. Ames & Massimo Mazzotti (eds.), Algorithmic modernity: mechanizing thought and action, 1500-2000. New York, NY: Oxford University Press.
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  9. Case for the Irreducibility of Geometry to Algebra†.Victor Pambuccian & Celia Schacht - 2022 - Philosophia Mathematica 30 (1):1-31.
    This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given (...)
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  10. The SuperHyperFunction and the Neutrosophic SuperHyperFunction (revisited again).Florentin Smarandache - 2022 - Neutrosophic Sets and Systems 49 (1):594-600.
    In this paper, one recalls the general definition of the SuperHyperAlgebra with its SuperHyperOperations and SuperHyperAxioms [2, 6]. Then one introduces for the first time the SuperHyperTopology and especially the SuperHyperFunction and Neutrosophic SuperHyperFunction. One gives a numerical example of a Neutro-SuperHyperGroup.
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  11. Introducción a la Super-Hiper-Álgebra y la Super-HiperÁlgebra Neutrosófica.Florentin Smarandache - 2022 - Neutrosophic Computing and Machine Learning 20 (1):1-6.
    In this article, the concepts of Nth Power Set of a Set, Super-Hyper-Oper-Operation, Super-Hyper-Axiom, SuperHyper-Algebra, and their corresponding Neutrosophic Super-Hyper-Oper-Operation, Neutrosophic Super-Hyper-Axiom and Neutrosophic Super-Hyper-Algebra are reviewed. In general, in any field of knowledge, really what are found are Super-HyperStructures (or more specifically Super-Hyper-Structures (m, n)).
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  12. Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra.Florentin Smarandache - 2022 - Journal of Algebraic Hyperstructures and Logical Algebras 3 (2):17-24.
    In this paper we recall our concepts of n th-Power Set of a Set, SuperHyperOperation, SuperHyperAxiom, SuperHyperAlgebra, and their corresponding Neutrosophic SuperHyperOperation, Neutrosophic SuperHyperAxiom and Neutrosophic SuperHyperAlgebra. In general, in any field of knowledge, one actually encounters SuperHyperStructures (or more accurately (m, n)- SuperHyperStructures).
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  13. On Some NeutroHyperstructures.Madeleine Al-Tahan, Bijan Davvaz, Florentin Smarandache & Osman Anis - 2021 - Symmetry 13 (4):1-12.
    Neutrosophy, the study of neutralities, is a new branch of Philosophy that has applications in many different fields of science. Inspired by the idea of Neutrosophy, Smarandache introduced NeutroAlgebraicStructures (or NeutroAlgebras) by allowing the partiality and indeterminacy to be included in the structures’ operations and/or axioms.
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  14. NeutroOrderedAlgebra: Applications to Semigroups.Madeleine Al-Tahan, Florentin Smarandache & Bijan Davvaz - 2021 - Neutrosophic Sets and Systems 39 (1):133-147.
    Starting with a partial order on a NeutroAlgebra, we get a NeutroStructure. The latter if it satisfies the conditions of NeutroOrder, it becomes a NeutroOrderedAlgebra. In this paper, we apply our new defined notion to semigroups by studying NeutroOrderedSemigroups. More precisely, we define some related terms like NeutrosOrderedSemigroup, NeutroOrderedIdeal, NeutroOrderedFilter, NeutroOrderedHomomorphism, etc., illustrate them via some examples, and study some of their properties.
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  15. Denseness results in the theory of algebraic fields.Sylvy Anscombe, Philip Dittmann & Arno Fehm - 2021 - Annals of Pure and Applied Logic 172 (8):102973.
    We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.
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  16. Cubic Polynomial for the Series of Consecutive Cubes under Alternating Signs.Leomarich Casinillo - 2021 - Inprime: Indonesian Journal of Pure and Applied Mathematics 3 (2):86-91.
    This paper aims to develop an elegant formula for the series of consecutive cubes of natural numbers under alternating signs. In addition, this paper investigates the formula under odd and even number of terms and discuss some important findings.
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  17. Some Results on a Generalized Version of Congruent Numbers.Leomarich Casinillo & Emily Casinillo - 2021 - Inprime: Indonesian Journal of Pure and Applied Mathematics 3 (1):1-6.
    This paper aims to construct a new formula that generates a generalized version of congruent numbers based on a generalized version of Pythagorean triples. Here, an elliptic curve equation is constructed from the derived generalized version of Pythagorean triples and congruent numbers and gives some new results.
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  18. Why Did Weyl Think That Emmy Noether Made Algebra the Eldorado of Axiomatics?Iulian D. Toader - 2021 - Hopos: The Journal of the International Society for the History of Philosophy of Science 11 (1):122-142.
    This paper argues that Noether's axiomatic method in algebra cannot be assimilated to Weyl's late view on axiomatics, for his acquiescence to a phenomenological epistemology of correctness led Weyl to resist Noether's principle of detachment.
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  19. 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 topological space form a Boolean (...)
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  20. Genealogy of Algorithms: Datafication as Transvaluation.Virgil W. Brower - 2020 - le Foucaldien 6 (1):1-43.
    This article investigates religious ideals persistent in the datafication of information society. Its nodal point is Thomas Bayes, after whom Laplace names the primal probability algorithm. It reconsiders their mathematical innovations with Laplace's providential deism and Bayes' singular theological treatise. Conceptions of divine justice one finds among probability theorists play no small part in the algorithmic data-mining and microtargeting of Cambridge Analytica. Theological traces within mathematical computation are emphasized as the vantage over large numbers shifts to weights beyond enumeration in (...)
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  21. An elementary transition to abstract mathematics.Gove W. Effinger - 2020 - Boca Raton: CRC Press, Taylor & Francis Group. Edited by Gary L. Mullen.
    An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role. The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part introduces (...)
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  22. 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 (eds.), 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 that inquisitive algebras determine Medvedev’s logic (...)
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  23. Transition to analysis with proof.Steven G. Krantz - 2018 - Boca Raton: CRC Press/Taylor & Francis Group.
    Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed by graduated exercises that (...)
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  24. Algebraic structures of neutrosophic triplets, neutrosophic duplets, or neutrosophic multisets.Florentin Smarandache, Xiaohong Zhang & Mumtaz Ali (eds.) - 2018 - Basel: MDPI.
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  25. Algebraic structures of neutrosophic triplets, neutrosophic duplets, or neutrosophic multisets. Volume I.Florentin Smarandache, Xiaohong Zhang & Mumtaz Ali - 2018 - Basel, Switzerland: MDPI. Edited by Florentin Smarandache, Xiaohong Zhang & Mumtaz Ali.
    The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued (...)
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  26. On the Interpretations of the History of Diophantine Analysis: A Comparative Study of Alternate Perspectives.Ioannis M. Vandoulakis - 2018 - Ganita Bharati 40 (3):115-152.
    Essay Review of “Les Arithmétiques de Diophante. Lecture historique et mathématique” by Roshdi Rashed and Christian Houzel, and Histoire de l’analyse diophantienne classique : d’Abū Kamil à Fermat by Roshdi Rashed.
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  27. ‘Chasing’ the diagram—the use of visualizations in algebraic reasoning.Silvia de Toffoli - 2017 - Review of Symbolic Logic 10 (1):158-186.
    The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...)
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  28. 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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  29. A Survey of Geometric Algebra and Geometric Calculus.Alan Macdonald - 2017 - Advances in Applied Clifford Algebras 27:853-891.
    The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
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  30. Philosophy of the Matrix.A. C. Paseau - 2017 - Philosophia Mathematica 25 (2):246-267.
    A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.
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  31. 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. Imagination in mathematics.Andrew Arana - 2016 - In Amy Kind (ed.), The Routledge Handbook of the Philosophy of Imagination. New York: Routledge. pp. 463-477.
    This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.
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  33. Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. Providence, RI: American Mathematical Society, 2014. Pp. xi + 441. ISBN 978-1-4704-1493-1. £79.95. [REVIEW]Michael J. Barany - 2016 - British Journal for the History of Science 49 (1):140-141.
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  34. Classical groups, derangements, and primes.Timothy C. Burness - 2016 - Cambridge: Cambridge University Press. Edited by Michael Giudici.
    A graduate-level introduction to finite classical groups featuring a comprehensive account of the conjugacy and geometry of elements of prime order.
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  35. Some classes of quasi-pseudo-MV algebras.Wenjuan Chen & Bijan Davvaz - 2016 - Logic Journal of the IGPL 24 (5).
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  36. A verifiable secret sharing scheme using non-abelian groups.Maggie E. Habeer - 2016 - In Delaram Kahrobaei, Bren Cavallo & David Garber (eds.), Algebra and computer science. Providence, Rhode Island: American Mathematical Society.
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  37. Algebra and computer science.Delaram Kahrobaei, Bren Cavallo & David Garber (eds.) - 2016 - Providence, Rhode Island: American Mathematical Society.
    This volume contains the proceedings of three special sessions: Algebra and Computer Science, held during the Joint AMS-EMS-SPM meeting in Porto, Portugal, June 10–13, 2015; Groups, Algorithms, and Cryptography, held during the Joint Mathematics Meeting in San Antonio, TX, January 10–13, 2015; and Applications of Algebra to Cryptography, held during the Joint AMS-Israel Mathematical Union meeting in Tel-Aviv, Israel, June 16–19, 2014. Papers contained in this volume address a wide range of topics, from theoretical aspects of algebra, namely group theory, (...)
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  38. Semigroups on MOD natural neutrosophic elements.Vasantha Kandasamy & B. W. - 2016 - Bruxelles, Belgium: EuropaNova. Edited by K. Ilanthenral & Florentin Smarandache.
    In this book the notion of semigroups under + is constructed using: the MOD natural neutrosophic integers, or MOD natural neutrosophic-neutrosophic numbers, or MOD natural neutrosophic finite complex modulo integer, or MOD natural neutrosophic dual number integers, or MOD natural neutrosophic special dual like number, or MOD natural neutrosophic special quasi dual numbers.
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  39. Knapsack and subset sum problems in nilpoint, polycyclic, and co-context-free groups.Daniel König, Markus Lohrey & George Zetzsche - 2016 - In Delaram Kahrobaei, Bren Cavallo & David Garber (eds.), Algebra and computer science. Providence, Rhode Island: American Mathematical Society.
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  40. On the Tits alternative for a class of finitely presented groups with a special focus on symbolic computations.Anja I. S. Moldenhauer, Gerhard Rosenberger & Kristina Rosenthal - 2016 - In Delaram Kahrobaei, Bren Cavallo & David Garber (eds.), Algebra and computer science. Providence, Rhode Island: American Mathematical Society.
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  41. Geometry of the conjugacy problem in lamplighter groups.Andew Salle - 2016 - In Delaram Kahrobaei, Bren Cavallo & David Garber (eds.), Algebra and computer science. Providence, Rhode Island: American Mathematical Society.
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  42. Development of abstract mathematical reasoning: the case of algebra.Ana Susac, Andreja Bubic, Andrija Vrbanc & Maja Planinic - 2016 - In Philippe Chassy & Wolfgang Grodd (eds.), Abstract mathematical cognition. [Lausanne, Switzerland]: Frontiers Media SA.
  43. A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups.Armin Weiss - 2016 - In Delaram Kahrobaei, Bren Cavallo & David Garber (eds.), Algebra and computer science. Providence, Rhode Island: American Mathematical Society.
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  44. The arithmetic of Z-numbers: theory and applications.Rafik A. Aliev - 2015 - Chennai: World Scientific. Edited by Oleg H. Huseynov, Rashad R. Aliyev & Akif A. Alizadeh.
    Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book is the first (...)
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  45. On ideals and congruences of distributive demi-p-algebras.T. S. Blyth, Jie Fang & Leibo Wang - 2015 - Studia Logica 103 (3):491-506.
    We identify the \-ideals of a distributive demi-pseudocomplemented algebra L as the kernels of the boolean congruences on L, and show that they form a complete Heyting algebra which is isomorphic to the interval \ of the congruence lattice of L where G is the Glivenko congruence. We also show that the notions of maximal \-ideal, prime \-ideal, and falsity ideal coincide.
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  46. Simple Logics for Basic Algebras.Ja̅nis Cı̅rulis - 2015 - Bulletin of the Section of Logic 44 (3/4):95-110.
    An MV-algebra is an algebra (A, ⊕, ¬, 0), where (A, ⊕, 0) is a commutative monoid and ¬ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also non-commutative) generalizations of MV-algebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in the sense of Blok and Pigozzi for two recent logical systems. Both are Hilbert-style systems, with implication and negation as the (...)
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  47. Representing Scott sets in algebraic settings.Alf Dolich, Julia F. Knight, Karen Lange & David Marker - 2015 - Archive for Mathematical Logic 54 (5-6):631-637.
    We prove that for every Scott set S there are S-saturated real closed fields and S-saturated models of Presburger arithmetic.
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  48. Meet-completions and ordered domain algebras.R. Egrot & Robin Hirsch - 2015 - Logic Journal of the IGPL 23 (4):584-600.
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  49. Complexity of equational theory of relational algebras with standard projection elements.Szabolcs Mikulás, Ildikó Sain & András Simon - 2015 - Synthese 192 (7):2159-2182.
    The class $$\mathsf{TPA}$$ TPA of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $$\mathsf{TPA}$$ TPA nor the first order theory of $$\mathsf{TPA}$$ TPA are decidable. Moreover, we show that the set of all equations valid in $$\mathsf{TPA}$$ TPA is exactly on the $$\Pi ^1_1$$ Π 1 1 level. We consider the class $$\mathsf{TPA}^-$$ (...)
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  50. Arithmetic, Mathematical Intuition, and Evidence.Richard Tieszen - 2015 - Inquiry: An Interdisciplinary Journal of Philosophy 58 (1):28-56.
    This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic . The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the (...)
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1 — 50 / 243