Numerical Semigroups
This monograph is the first devoted exclusively to the development of the theory of numerical semigroups. In this concise, self-contained text, graduate students and researchers will benefit from this broad exposition of the topic. Key features of "Numeri
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Numerical Semigroups
20
NUMERICAL SEMIGROUPS
Developments in Mathematics VOLUME 20 Series Editor: Krishnaswami Alladi, University of Florida, U.S.A.
NUMERICAL SEMIGROUPS
By
J.C. ROSALES University of Granada, Spain ´ P.A. GARC´IA-SANCHEZ University of Granada, Spain
123
J.C. Rosales Department of Algebra Faculty of Sciences University of Granada Campus University Fuentenueva 18071 Granada Spain [email protected]
P.A. Garc´ıa-S´anchez Department of Algebra Faculty of Sciences University of Granada Campus University Fuentenueva 18071 Granada Spain [email protected]
ISSN 1389-2177 ISBN 978-1-4419-0159-0 e-ISBN 978-1-4419-0160-6 DOI 10.1007/978-1-4419-0160-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009934163 Mathematics Subject Classification (2000): 20M14, 13H10, 11DXX 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 is part of Springer Science+Business Media (www.springer.com)
Para Loly, Patricia y Carlos Para Mar´ıa y Alba
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
Notable elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 Monoids and monoid homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Multiplicity and embedding dimension . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Frobenius number and genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Pseudo-Frobenius numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2
Numerical semigroups with maximal embedding dimension . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Arf numerical semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Saturated numerical semigroups . . . . . . . . . . . . . . .
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