Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science
Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

Author: Janusz Czelakowski

Publisher: Springer

Published: 2018-03-20

Total Pages: 476

ISBN-13: 331974772X

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This book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life. It reflects Pigozzi's contribution to the rise and development of areas such as abstract algebraic logic (AAL), universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically ordered facts relating to the development of the disciplines he contributed to, especially abstract algebraic logic. The book offers valuable source material for historians of science, especially those interested in history of mathematics and logic.

Universal Algebra for Computer Scientists
Universal Algebra for Computer Scientists

Author: Wolfgang Wechler

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 345

ISBN-13: 3642767710

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A new model-theoretic approach to universal algebra is offered in this book. Written for computer scientists, it presents a systematic development of the methods and results of universal algebra that are useful in a variety of applications in computer science. The notation is simple and the concepts are clearly presented. The book concerns the algebraic characterization of axiomatic classes of algebras (equational, implicational, and universal Horn classes) by closure operators generalizing the famous Birkhoff Variety Theorem, and the algebraic characterization of the related theories. The book also presents a thorough study of term rewriting systems. Besides basic notions, the Knuth-Bendix completion procedure and termination proof methods are considered. A third main topic is that of fixpoint techniques and complete ordered algebras. Algebraic specifications of abstract data types and algebraic semantics of recursive program schemes are treated as applications. The book is self-contained and suitable both as a textbook for graduate courses and as a reference for researchers.

A Course in Universal Algebra
A Course in Universal Algebra

Author: S. Burris

Publisher: Springer

Published: 2011-10-21

Total Pages: 276

ISBN-13: 9781461381327

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Universal algebra has enjoyed a particularly explosive growth in the last twenty years, and a student entering the subject now will find a bewildering amount of material to digest. This text is not intended to be encyclopedic; rather, a few themes central to universal algebra have been developed sufficiently to bring the reader to the brink of current research. The choice of topics most certainly reflects the authors' interests. Chapter I contains a brief but substantial introduction to lattices, and to the close connection between complete lattices and closure operators. In particular, everything necessary for the subsequent study of congruence lattices is included. Chapter II develops the most general and fundamental notions of uni versal algebra-these include the results that apply to all types of algebras, such as the homomorphism and isomorphism theorems. Free algebras are discussed in great detail-we use them to derive the existence of simple algebras, the rules of equational logic, and the important Mal'cev conditions. We introduce the notion of classifying a variety by properties of (the lattices of) congruences on members of the variety. Also, the center of an algebra is defined and used to characterize modules (up to polynomial equivalence). In Chapter III we show how neatly two famous results-the refutation of Euler's conjecture on orthogonal Latin squares and Kleene's character ization of languages accepted by finite automata-can be presented using universal algebra. We predict that such "applied universal algebra" will become much more prominent.

Algebraic Methods in Philosophical Logic
Algebraic Methods in Philosophical Logic

Author: J. Michael Dunn

Publisher: OUP Oxford

Published: 2001-06-28

Total Pages: 490

ISBN-13: 0191589225

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This comprehensive text demonstrates how various notions of logic can be viewed as notions of universal algebra. It is aimed primarily for logisticians in mathematics, philosophy, computer science and linguistics with an interest in algebraic logic, but is also accessible to those from a non-logistics background. It is suitable for researchers, graduates and advanced undergraduates who have an introductory knowledge of algebraic logic providing more advanced concepts, as well as more theoretical aspects. The main theme is that standard algebraic results (representations) translate into standard logical results (completeness). Other themes involve identification of a class of algebras appropriate for classical and non-classical logic studies, including: gaggles, distributoids, partial- gaggles, and tonoids. An imporatant sub title is that logic is fundamentally information based, with its main elements being propositions, that can be understood as sets of information states. Logics are considered in various senses e.g. systems of theorems, consequence relations and, symmetric consequence relations.

Foundations of Algebraic Specification and Formal Software Development
Foundations of Algebraic Specification and Formal Software Development

Author: Donald Sannella

Publisher: Springer Science & Business Media

Published: 2012-01-05

Total Pages: 594

ISBN-13: 3642173365

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This book provides foundations for software specification and formal software development from the perspective of work on algebraic specification, concentrating on developing basic concepts and studying their fundamental properties. These foundations are built on a solid mathematical basis, using elements of universal algebra, category theory and logic, and this mathematical toolbox provides a convenient language for precisely formulating the concepts involved in software specification and development. Once formally defined, these notions become subject to mathematical investigation, and this interplay between mathematics and software engineering yields results that are mathematically interesting, conceptually revealing, and practically useful. The theory presented by the authors has its origins in work on algebraic specifications that started in the early 1970s, and their treatment is comprehensive. This book contains five kinds of material: the requisite mathematical foundations; traditional algebraic specifications; elements of the theory of institutions; formal specification and development; and proof methods. While the book is self-contained, mathematical maturity and familiarity with the problems of software engineering is required; and in the examples that directly relate to programming, the authors assume acquaintance with the concepts of functional programming. The book will be of value to researchers and advanced graduate students in the areas of programming and theoretical computer science.

Universal Algebra and Applications in Theoretical Computer Science
Universal Algebra and Applications in Theoretical Computer Science

Author: Klaus Denecke

Publisher: CRC Press

Published: 2018-10-03

Total Pages: 396

ISBN-13: 1482285835

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Over the past 20 years, the emergence of clone theory, hyperequational theory, commutator theory and tame congruence theory has led to a growth of universal algebra both in richness and in applications, especially in computer science. Yet most of the classic books on the subject are long out of print and, to date, no other book has integrated these theories with the long-established work that supports them. Universal Algebra and Applications in Theoretical Computer Science introduces the basic concepts of universal algebra and surveys some of the newer developments in the field. The first half of the book provides a solid grounding in the core material. A leisurely pace, careful exposition, numerous examples, and exercises combine to form an introduction to the subject ideal for beginning graduate students or researchers from other areas. The second half of the book focuses on applications in theoretical computer science and advanced topics, including Mal'cev conditions, tame congruence theory, clones, and commutators. The impact of the advances in universal algebra on computer science is just beginning to be realized, and the field will undoubtedly continue to grow and mature. Universal Algebra and Applications in Theoretical Computer Science forms an outstanding text and offers a unique opportunity to build the foundation needed for further developments in its theory and in its computer science applications.

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.