The Equationally-Defined Commutator A Study in Equational Logic and
This monograph introduces and explores the notions of a commutator equation and the equationally-defined commutator from the perspective of abstract algebraic logic. An account of the commutator operation associated with equational deductive systems
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The EquationallyDefined Commutator A Study in Equational Logic and Algebra
Janusz Czelakowski
The Equationally-Defined Commutator A Study in Equational Logic and Algebra
Janusz Czelakowski Institute of Mathematics and Informatics Opole University Opole, Poland
ISBN 978-3-319-21199-2 ISBN 978-3-319-21200-5 (eBook) DOI 10.1007/978-3-319-21200-5 Library of Congress Control Number: 2015947504 Mathematics Subject Classification (2010): 08-02, 03C05, 03G27, 06C05, 08A30, 08A35, 08B05, 08B10, 08C15 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
Synopsis
The purpose of this book is to present in a uniform way commutator theory for universal algebra. We are interested in the logical perspective of the research— emphasis is put on an analysis of the interconnections holding between the commutator and equational logic. This book thus qualifies as belonging to abstract algebraic logic (AAL), the area of research that explores to a large extent the methods provided by the general theory of deductive systems.1 The notion of a commutator equation2 is introduced, and it plays a central role in the theory to be expounded. This book is therefore concerned with the meanings the term “commutator equation” receives in the models provided by theory and clarifies the contexts in which these meanings occur. Purely syntactic aspects of the theory of the commutator are underlined. This book is mainly addressed to algebraists and logicians.
1 From the viewpoint of AAL, universal algebra is the study of (reduced) models of equational logics. The latter are defined as structural and finitary strengthenings of the basic equational Birkhoff’s logic. 2 The term “commutator equation” denotes in this book a certain equation from the first-order language of a
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