Function Algebras on Finite Sets A Basic Course on Many-Valued Logic
Functions which are defined on finite sets occur in almost all fields of mathematics. For more than 80 years algebras whose universes are such functions (so-called function algebras), have been intensively studied. This book gives a broad introduction to
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Dietlinde Lau
Function Algebras on Finite Sets A Basic Course on Many-Valued Logic and Clone Theory
With 42 Figures and 46 Tables
123
Dietlinde Lau Institute for Mathematics University of Rostock Universitätsplatz 1 18055 Rostock, Germany e-mail: [email protected]
Library of Congress Control Number: 2006929534
Mathematics Subject Classification (2000): 03B50, 08Axx, 08A40, 08A30, 08A05, 06A15 ISSN 1439-7382 ISBN-10 3-540-36022-0 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-36022-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, 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. Typesetting by the author using a Springer TEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper
44/3100YL - 5 4 3 2 1 0
To my mother, Brigitte Lau
Preface
Functions (or operations), which are defined on finite sets, occur in almost all fields of mathematics. For more than 80 years, algebras (so-called function algebras), whose universes are such functions, have been studied. Particularly in Mathematical Logic, in Universal Algebra (more precise in the Clone Theory), and in parts of Computer Science, certain knowledge about these algebras are subject of the fundamental knowledge. Currently only one book has been published about function algebras, apart from certain monographs or dissertations of specific themes, survey articles and books that contain sections about function algebras or clones. This book has been written by R. P¨ oschel und L. A. Kaluˇznin in the German language and gives a very good overview about the results achieved up to 1979. During the last 26 years, many new results have been obtained; however, a new book about function algebras is overdue. The aim of the present book is to introduce the reader to the theory of function algebras and to give the latest state of research for some selected fields. The author would like to acquaint the reader with proof of the fundamental theorems and the different proof methods, to enable research in the field of the function algebras. This book is self-contai
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