Algebraic Theories
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Author | : E.G. Manes |
Publisher | : Springer Science & Business Media |
Total Pages | : 364 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461298601 |
In the past decade, category theory has widened its scope and now inter acts with many areas of mathematics. This book develops some of the interactions between universal algebra and category theory as well as some of the resulting applications. We begin with an exposition of equationally defineable classes from the point of view of "algebraic theories," but without the use of category theory. This serves to motivate the general treatment of algebraic theories in a category, which is the central concern of the book. (No category theory is presumed; rather, an independent treatment is provided by the second chap ter.) Applications abound throughout the text and exercises and in the final chapter in which we pursue problems originating in topological dynamics and in automata theory. This book is a natural outgrowth of the ideas of a small group of mathe maticians, many of whom were in residence at the Forschungsinstitut für Mathematik of the Eidgenössische Technische Hochschule in Zürich, Switzerland during the academic year 1966-67. It was in this stimulating atmosphere that the author wrote his doctoral dissertation. The "Zürich School," then, was Michael Barr, Jon Beck, John Gray, Bill Lawvere, Fred Linton, and Myles Tierney (who were there) and (at least) Harry Appelgate, Sammy Eilenberg, John Isbell, and Saunders Mac Lane (whose spiritual presence was tangible.) I am grateful to the National Science Foundation who provided support, under grants GJ 35759 and OCR 72-03733 A01, while I wrote this book.
Author | : J. Adámek |
Publisher | : Cambridge University Press |
Total Pages | : 268 |
Release | : 2010-11-18 |
Genre | : Mathematics |
ISBN | : 9780521119221 |
Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.
Author | : Leonard Dickson |
Publisher | : Courier Corporation |
Total Pages | : 241 |
Release | : 2014-03-05 |
Genre | : Mathematics |
ISBN | : 048615520X |
This in-depth introduction to classical topics in higher algebra provides rigorous, detailed proofs for its explorations of some of mathematics' most significant concepts, including matrices, invariants, and groups. Algebraic Theories studies all of the important theories; its extensive offerings range from the foundations of higher algebra and the Galois theory of algebraic equations to finite linear groups (including Klein's "icosahedron" and the theory of equations of the fifth degree) and algebraic invariants. The full treatment includes matrices, linear transformations, elementary divisors and invariant factors, and quadratic, bilinear, and Hermitian forms, both singly and in pairs. The results are classical, with due attention to issues of rationality. Elementary divisors and invariant factors receive simple, natural introductions in connection with the classical form and a rational, canonical form of linear transformations. All topics are developed with a remarkable lucidity and discussed in close connection with their most frequent mathematical applications.
Author | : Edgar Dehn |
Publisher | : Courier Corporation |
Total Pages | : 225 |
Release | : 2012-09-05 |
Genre | : Mathematics |
ISBN | : 0486155102 |
Focusing on basics of algebraic theory, this text presents detailed explanations of integral functions, permutations, and groups as well as Lagrange and Galois theory. Many numerical examples with complete solutions. 1930 edition.
Author | : J. Adámek |
Publisher | : Cambridge University Press |
Total Pages | : 269 |
Release | : 2010-11-18 |
Genre | : Mathematics |
ISBN | : 1139491881 |
Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.
Author | : Paulo Ribenboim |
Publisher | : Springer Science & Business Media |
Total Pages | : 676 |
Release | : 2013-11-11 |
Genre | : Mathematics |
ISBN | : 0387216901 |
The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.
Author | : Richard Dedekind |
Publisher | : Cambridge University Press |
Total Pages | : 170 |
Release | : 1996-09-28 |
Genre | : Mathematics |
ISBN | : 0521565189 |
A translation of a classic work by one of the truly great figures of mathematics.
Author | : Adrian Albert A |
Publisher | : Legare Street Press |
Total Pages | : 0 |
Release | : 2022-10-27 |
Genre | : |
ISBN | : 9781018171746 |
This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Author | : Leonard Eugene Dickson |
Publisher | : |
Total Pages | : 296 |
Release | : 1926 |
Genre | : Algebra |
ISBN | : |
Author | : Régine Douady |
Publisher | : Springer Nature |
Total Pages | : 462 |
Release | : 2020-07-13 |
Genre | : Mathematics |
ISBN | : 3030327965 |
Galois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings. This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory.