Commutative Semigroups
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Advances in Mathematics VOLUME2
Series Editor: J. Szep, Budapest University of Economics, Hungary Advisory Board: G. Erjaee, Shiraz University, Iran W. Fouche, University of South Africa, South Africa P. Grillet, Tulane University, U.S.A. H.J. Hoehnke, Germany F. Szidarovszky, University of Arizona, U.S.A.
P. Zecca, Universita di Firenze, Italy
The titles published in this series are listed at the end of this volume.
Commutative Semigroups
by
P.A. GRILLET Tulane University, New Orleans, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA. B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4857-1 DOI 10.1007/978-1-4757-3389-1
ISBN 978-1-4757-3389-1 (eBook)
Printed on acid-free paper
All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by K.luwer Academic Publishers in 2001 Softcover reprint of the hardcover 1st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
A ma mere a qui je dois tout
CONTENTS Preface
. . . . . . . . . . . . . . • • . . XI
General structure theory Chapter I. Elementary properties 1. First results . . . . . 2. Homomorphisms and congruences 3. Ideals . . . . . . . . . . 4. Divisibility . . . . . . . 5. Free commutative semigroups 6. Presentations . . . . . .
I 6 13 I7 20 25
Chapter II. Cancellative semigroups I. Semigroups of fractions 2. Universal groups 3. Cancellative semigroups 4. Numerical semigroups 5. General structure 6. Faces . . . . 7. Free embedding 8. Krull monoids
29 29 32 36 39 44 50 54 57
Chapter III. Semilattice decompositions I. General results . . . . . . . 2. Clifford semigroups . . . . . 3. Complete archimedean semigroups 4. N-semigroups . . . . . . . . 5. Subcompiete archimedean semigroups 6. Power-joined semigroups . . Chapter IV. Subdirect decompositions I. Subdirect products 2. Separative semigroups 3. N ilsemigroups 4. Ponizovsky decompositions 5. Elementary semigroups
69 69 73 78 82
107 112
Chapter V. Group coextensions . 1. Dividing by J{ . . . . 2. SchOtzenberger functors 3. Coextensions . . . . . 4. Group coextensions 5. Subdirectly irreducible semigroups
115 115 117 120 I25 133
vii
86 90 95 95
IOI I04
viii
CONTENTS.
Chapter VI. Finitely generated semigroups 1. Redei's Theorem . . . . 2. Subdirect decompositions . 3. Subelementary semigroups 4. The Completion Theorem 5. Irreducible semigroups . . 6. Archimedean semigroups . 7. The a.c.c. on subsemigroups
141 141 144 148 149 154 158 160
Chapter VII. Subcomplete semigroups 1. Completions . . . . . . . 2. Ponizovsky families . . . . 3. Another Completion Theorem 4. Properties . . . . . . 5. Schutzenberger functors
165 165 169 173 179 181
Chapter VIII. Other results 1. Examples . . . . 2. Products and subsets 3. Homomorphisms and congruences . . . . . . 4. Other topic
