# Algebra

Edited by Nemi Boris Pelgrom (Ludwig Maximilians Universität, München)
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1. 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. 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. 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. 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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5. 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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6. 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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7. 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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8. 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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9. 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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10. 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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11. 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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12. 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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13. 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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14. 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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15. ‘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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16. 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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17. 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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18. 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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19. SOFT NEUTROSOPHIC ALGEBRAIC STRUCTURES AND THEIR GENERALIZATION, Vol. 1.Florentin Smarandache, Mumtaz Ali & Muhammad Shabir - 2014 - Columbus, OH, USA: Educational Publisher.
In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft (...)

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20. SOFT NEUTROSOPHIC ALGEBRAIC STRUCTURES AND THEIR GENERALIZATION, Vol. 2.Florentin Smarandache, Mumtaz Ali & Muhammad Shabir - 2014 - Columbus, OH, USA: Educational Publisher.
In this book we define some new notions of soft neutrosophic algebraic structures over neutrosophic algebraic structures. We define some different soft neutrosophic algebraic structures but the main motivation is two-fold. Firstly the classes of soft neutrosophic group ring and soft neutrosophic semigroup ring defined in this book is basically the generalization of two classes of rings: neutrosophic group rings and neutrosophic semigroup rings. These soft neutrosophic group rings and soft neutrosophic semigroup rings are defined over neutrosophic group rings and (...)

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21. Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics.Jean-Pierre Marquis - 2013 - Synthese 190 (12):2141-2164.
In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...)

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22. Notes on Groups and Geometry, 1978-1986.Steven H. Cullinane - 2012 - Internet Archive.
Typewritten notes on groups and geometry.

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23. Algebraic symbolism in medieval Arabic algebra.Jeffrey A. Oaks - 2012 - Philosophica 87 (4):27-83.

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24. Neutrosophic Super Matrices and Quasi Super Matrices.Florentin Smarandache & W. B. Vasantha Kandasamy - 2012 - Columbus, OH, USA: Zip Publishing.
In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution.

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25. Algebraic Structures using Super Interval Matrices.W. B. Vasantha Kandasamy & Florentin Smarandache - 2011 - Columbus, OH, USA: Educational Publisher.
In this book authors for the first time introduce the notion of super interval matrices using special intervals. The advantage of using super interval matrices is that one can build only one vector space using m × n interval matrices, but in case of super interval matrices we can have several such spaces depending on the partition on the interval matrix.

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26. Super Special Codes using Super Matrices.W. B. Vasantha Kandasamy, Florentin Smarandache & K. Ilanthenral - 2010 - Stockholm, Sweden: Svenska fysikarkivet.
The new classes of super special codes are constructed in this book using the specially constructed super special vector spaces. These codes mainly use the super matrices. These codes can be realized as a special type of concatenated codes.

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27. Two conjectures on the arithmetic in ℝ and ℂ†.Apoloniusz Tyszka - 2010 - Mathematical Logic Quarterly 56 (2):175-184.
Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ, then S has a real solution which consists of numbers whose absolute values belong to (...)

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28. Superbimatrices and Their Generalizations.W. B. Vasantha Kandasamy & Florentin Smarandache - 2009 - Slatina, Romania: CuArt.
The systematic study of supermatrices and super linear algebra has been carried out in 2008. These new algebraic structures find their applications in fuzzy models, Leontief economic models and data-storage in computers.

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29. Super Linear Algebra.W. B. Vasantha Kandasamy & Florentin Smarandache - 2008 - Ann Arbor, MI, USA: ProQuest Information & Learning.
In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such (...)