The q-theory of Finite Semigroups
Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic te
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John Rhodes . Benjamin Steinberg
The q-theory of Finite Semigroups
123
John Rhodes University of California Department of Mathematics 1000 Centennial Dr. Berkeley CA 94720-3840 USA [email protected]
ISSN: 1439-7382 ISBN: 978-0-387-09780-0 DOI 10.1007/978-0-387-09781-7
Benjamin Steinberg Cereleton University School of Mathematics and Statistics 1125 Colonel by Drive Ottawa ON K1S 5B6 Canada [email protected]
e-ISBN: 978-0-387-09781-7
Library of Congress Control Number: 2008939929
Mathematics Subject Classification (2000): 20M07, 20M35, 20F20, 20F32, 20A15, 20A26, 16Y60, 06B35, 06E15, 06A15 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
In memory of Bret Tilson
Preface
When people are trying to learn mathematics for the purpose of research, they usually start at the forefront and then go backwards as needed in order to understand the results more fully. Yet with a mathematics book, it is not uncommon for people to start on page one and read onwards. This book is a research manuscript, and we heartily encourage the reader to delve in, read what is of interest, and go back as necessary. We hope that the material at the end of the book — the charts, tables and indices — will make this easier going. Not many books have appeared in recent years dedicated to state-of-theart Finite Semigroup Theory. There was the famous “Arbib” book in the 1960s [171] containing the lectures of Kenneth Krohn, John Rhodes and Bret Tilson. Samuel Eilenberg’s treatise [85], with two chapters by Tilson [362,363], appeared more than 30 years ago. It revolutionized semigroup theory with the introduction of pseudovarieties of semigroups and varieties of languages. However, the most recent book on the subject is that of Jorge Almeida [7], which was originally published in 1992! Almeida’s book made profinite methods in semigroup theory accessible. Howard Straubing’s 1994 book [346] does touch on semigroup theory, but it is more concerned with applications to Computer Science than with semigroups themselves. This volume is intended both to introduce a quantized version of Eilenberg’s theory, by going to operators on pseudovarieties and relational morphisms, and to fill in some of the vac
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