Algebras, Lattices, Varieties

Algebras, Lattices, Varieties

Author: Ralph S. Freese

Publisher: American Mathematical Society

Published: 2022-10-28

Total Pages: 496

ISBN-13: 1470467976

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This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.


Book Synopsis Algebras, Lattices, Varieties by : Ralph S. Freese

Download or read book Algebras, Lattices, Varieties written by Ralph S. Freese and published by American Mathematical Society. This book was released on 2022-10-28 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.

Algebras, Lattices, Varieties

Algebras, Lattices, Varieties

Author: Ralph N. McKenzie

Publisher: American Mathematical Society

Published: 2018-07-09

Total Pages: 386

ISBN-13: 1470442957

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This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text.


Book Synopsis Algebras, Lattices, Varieties by : Ralph N. McKenzie

Download or read book Algebras, Lattices, Varieties written by Ralph N. McKenzie and published by American Mathematical Society. This book was released on 2018-07-09 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text.

Algebras, lattices, varieties

Algebras, lattices, varieties

Author: Ralph N. MacKenzie

Publisher:

Published: 1987

Total Pages: 0

ISBN-13:

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Book Synopsis Algebras, lattices, varieties by : Ralph N. MacKenzie

Download or read book Algebras, lattices, varieties written by Ralph N. MacKenzie and published by . This book was released on 1987 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Varieties of Lattices

Varieties of Lattices

Author: Peter Jipsen

Publisher: Springer

Published: 2006-11-15

Total Pages: 171

ISBN-13: 3540475141

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The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.


Book Synopsis Varieties of Lattices by : Peter Jipsen

Download or read book Varieties of Lattices written by Peter Jipsen and published by Springer. This book was released on 2006-11-15 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.

M-Solid Varieties of Algebras

M-Solid Varieties of Algebras

Author: Jörg Koppitz

Publisher: Springer Science & Business Media

Published: 2006-02-10

Total Pages: 364

ISBN-13: 9780387308043

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A complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on solid varieties of semirings and semigroups. The book aims to develop the theory of solid varieties as a system of mathematical discourse that is applicable in several concrete situations. A unique feature of this book is the use of Galois connections to integrate different topics.


Book Synopsis M-Solid Varieties of Algebras by : Jörg Koppitz

Download or read book M-Solid Varieties of Algebras written by Jörg Koppitz and published by Springer Science & Business Media. This book was released on 2006-02-10 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: A complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on solid varieties of semirings and semigroups. The book aims to develop the theory of solid varieties as a system of mathematical discourse that is applicable in several concrete situations. A unique feature of this book is the use of Galois connections to integrate different topics.

Algebras, Lattices, Varieties

Algebras, Lattices, Varieties

Author: Ralph S. Freese

Publisher: American Mathematical Society

Published: 2022-11-03

Total Pages: 451

ISBN-13: 1470467984

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This book is the third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.


Book Synopsis Algebras, Lattices, Varieties by : Ralph S. Freese

Download or read book Algebras, Lattices, Varieties written by Ralph S. Freese and published by American Mathematical Society. This book was released on 2022-11-03 with total page 451 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.

The Lattice of Interpretability Types of Varieties

The Lattice of Interpretability Types of Varieties

Author: Octavio Carlos García

Publisher: American Mathematical Soc.

Published: 1984

Total Pages: 133

ISBN-13: 0821823086

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We investigate the lattice, invented by W. D. Neumann in 1974, formed by the class of all varieties under the quasi-ordering "[script]V is interpretable in [script]W." The lattice is found to be non-modular and a proper class. Various familiar varieties are found to be [logical conjunction symbol {up arrow}]-irreducible (or prime) and various filters (especially Mal'tsev classes) are found to be indecomposable (or prime). Many familiar varieties are found to be inequivalent in the lattice, using a new technique of SIN algebras. Seven figures are included which document the known relationships between some sixty known or easily describable varieties and varietal families.


Book Synopsis The Lattice of Interpretability Types of Varieties by : Octavio Carlos García

Download or read book The Lattice of Interpretability Types of Varieties written by Octavio Carlos García and published by American Mathematical Soc.. This book was released on 1984 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: We investigate the lattice, invented by W. D. Neumann in 1974, formed by the class of all varieties under the quasi-ordering "[script]V is interpretable in [script]W." The lattice is found to be non-modular and a proper class. Various familiar varieties are found to be [logical conjunction symbol {up arrow}]-irreducible (or prime) and various filters (especially Mal'tsev classes) are found to be indecomposable (or prime). Many familiar varieties are found to be inequivalent in the lattice, using a new technique of SIN algebras. Seven figures are included which document the known relationships between some sixty known or easily describable varieties and varietal families.

The Structure of Finite Algebras

The Structure of Finite Algebras

Author: David Charles Hobby

Publisher:

Published: 1988

Total Pages: 220

ISBN-13:

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The utility of congruence lattices in revealing the structure of general algebras has been recognized since Garrett Birkhoff's pioneering work in the 1930s and 1940s. However, the results presented in this book are of very recent origin: most of them were developed in 1983. The main discovery presented here is that the lattice of congruences of a finite algebra is deeply connected to the structure of that algebra. The theory reveals a sharp division of locally finite varieties of algebras into six interesting new families, each of which is characterized by the behavior of congruences in the algebras. The authors use the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well. The authors begin with a straightforward and complete development of basic tame congruence theory, a topic that offers great promise for a wide variety of investigations. They then move beyond the consideration of individual algebras to a study of locally finite varieties. A list of open problems closes the work.


Book Synopsis The Structure of Finite Algebras by : David Charles Hobby

Download or read book The Structure of Finite Algebras written by David Charles Hobby and published by . This book was released on 1988 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The utility of congruence lattices in revealing the structure of general algebras has been recognized since Garrett Birkhoff's pioneering work in the 1930s and 1940s. However, the results presented in this book are of very recent origin: most of them were developed in 1983. The main discovery presented here is that the lattice of congruences of a finite algebra is deeply connected to the structure of that algebra. The theory reveals a sharp division of locally finite varieties of algebras into six interesting new families, each of which is characterized by the behavior of congruences in the algebras. The authors use the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well. The authors begin with a straightforward and complete development of basic tame congruence theory, a topic that offers great promise for a wide variety of investigations. They then move beyond the consideration of individual algebras to a study of locally finite varieties. A list of open problems closes the work.

Lectures in Universal Algebra

Lectures in Universal Algebra

Author: L. Szabó

Publisher: Elsevier

Published: 2016-01-22

Total Pages: 657

ISBN-13: 1483295400

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These 34 papers cover topics ranging from various problems on varieties and other classes of algebras including categorical aspects and duality theory to the structure of finite algebras and clones on finite (or infinite) sets.As well as survey articles by invited speakers, the papers contain full proofs of new results not published elsewhere. The volume ends with a list of problems.


Book Synopsis Lectures in Universal Algebra by : L. Szabó

Download or read book Lectures in Universal Algebra written by L. Szabó and published by Elsevier. This book was released on 2016-01-22 with total page 657 pages. Available in PDF, EPUB and Kindle. Book excerpt: These 34 papers cover topics ranging from various problems on varieties and other classes of algebras including categorical aspects and duality theory to the structure of finite algebras and clones on finite (or infinite) sets.As well as survey articles by invited speakers, the papers contain full proofs of new results not published elsewhere. The volume ends with a list of problems.

Ordered Sets and Lattices II

Ordered Sets and Lattices II

Author:

Publisher: American Mathematical Soc.

Published:

Total Pages: 262

ISBN-13: 9780821895887

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This indispensable reference source contains a wealth of information on lattice theory. The book presents a survey of virtually everything published in the fields of partially ordered sets, semilattices, lattices, and Boolean algebras that was reviewed in Referativnyi Zhurnal Matematika from mid-1982 to the end of 1985. A continuation of a previous volume (the English translation of which was published by the AMS in 1989, as volume 141 in Translations - Series 2), this comprehensive work contains more than 2200 references. Many of the papers covered here were originally published in virtually inaccessible places. The compilation of the volume was directed by Milan Kolibiar of Comenius University at Bratislava and Lev A. Skornyakov of Moscow University. Of interest to mathematicians, as well as to philosophers and computer scientists in certain areas, this unique compendium is a must for any mathematical library.


Book Synopsis Ordered Sets and Lattices II by :

Download or read book Ordered Sets and Lattices II written by and published by American Mathematical Soc.. This book was released on with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: This indispensable reference source contains a wealth of information on lattice theory. The book presents a survey of virtually everything published in the fields of partially ordered sets, semilattices, lattices, and Boolean algebras that was reviewed in Referativnyi Zhurnal Matematika from mid-1982 to the end of 1985. A continuation of a previous volume (the English translation of which was published by the AMS in 1989, as volume 141 in Translations - Series 2), this comprehensive work contains more than 2200 references. Many of the papers covered here were originally published in virtually inaccessible places. The compilation of the volume was directed by Milan Kolibiar of Comenius University at Bratislava and Lev A. Skornyakov of Moscow University. Of interest to mathematicians, as well as to philosophers and computer scientists in certain areas, this unique compendium is a must for any mathematical library.