Semigroups in Complete Lattices Quantales, Modules and Related Topic
This monograph provides a modern introduction to the theory of quantales.First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer scien
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Patrik Eklund Javier Gutiérrez García Ulrich Höhle Jari Kortelainen
Semigroups in Complete Lattices Quantales, Modules and Related Topics
Developments in Mathematics Volume 54
Series editors Krishnaswami Alladi, Gainesville, USA Pham Huu Tiep, Piscataway, USA Loring W. Tu, Medford, USA
More information about this series at http://www.springer.com/series/5834
Patrik Eklund Javier Gutiérrez García Ulrich Höhle Jari Kortelainen •
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Semigroups in Complete Lattices Quantales, Modules and Related Topics
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Patrik Eklund Department of Computing Science Umeå University Umeå Sweden Javier Gutiérrez García Department of Mathematics University of the Basque Country UPV/EHU Bilbao Spain
Ulrich Höhle School of Mathematics and Natural Sciences University of Wuppertal Wuppertal Germany Jari Kortelainen Department of Information Technology South-Eastern Finland University of Applied Sciences Mikkeli Finland
ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-78947-7 ISBN 978-3-319-78948-4 (eBook) https://doi.org/10.1007/978-3-319-78948-4 Library of Congress Control Number: 2018938638 Mathematics Subject Classification (2010): 06F07, 18C20, 18D15, 46L05 © Springer International Publishing AG, part of Springer Nature 2018 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To Aija, Ana, Felix, Heidi-Tuulia and Kesia
Foreword
This book is about quantales, lattice-ordered semigroups structured in such a way that they generalize the locales of point-free topology and various multiplicative lattices of ideals from the theories of rings and C -algebras. The formal notion of a quantale—a complete lattice equipped w
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