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  • Book
  • © 2009

Numerical Semigroups

  • First monograph devoted exclusively to the study of numerical semigroups

  • Presents various applications of numerical semigroups including number theory, coding theory, algebraic geometry, and others

  • The text is self contained

  • Presents content ranging from the basics to open research problems

  • Accessible to any first year graduate student

  • Includes exercises that invite the reader to prove results published in recent mathematical journals

  • Includes supplementary material: sn.pub/extras

Part of the book series: Developments in Mathematics (DEVM, volume 20)

Buying options

eBook USD 84.99
Price excludes VAT (USA)
  • ISBN: 978-1-4419-0160-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 109.99
Price excludes VAT (USA)
Hardcover Book USD 109.99
Price excludes VAT (USA)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-ix
  2. Introduction

    • J.C. Rosales, P.A. García-Sánchez
    Pages 1-3
  3. Notable elements

    • J.C. Rosales, P.A. García-Sánchez
    Pages 5-18
  4. Numerical semigroups with maximal embedding dimension

    • J.C. Rosales, P.A. García-Sánchez
    Pages 19-32
  5. Irreducible numerical semigroups

    • J.C. Rosales, P.A. García-Sánchez
    Pages 33-55
  6. Proportionally modular numerical semigroups

    • J.C. Rosales, P.A. García-Sánchez
    Pages 57-76
  7. The quotient of a numerical semigroup by a positive integer

    • J.C. Rosales, P.A. García-Sánchez
    Pages 77-90
  8. Presentations of a numerical semigroup

    • J.C. Rosales, P.A. García-Sánchez
    Pages 105-122
  9. The gluing of numerical semigroups

    • J.C. Rosales, P.A. García-Sánchez
    Pages 123-136
  10. Numerical semigroups with embedding dimension three

    • J.C. Rosales, P.A. García-Sánchez
    Pages 137-154
  11. The structure of a numerical semigroup

    • J.C. Rosales, P.A. García-Sánchez
    Pages 155-169
  12. Back Matter

    Pages 171-181

About this book

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 "Numerical Semigroups" include:

- Content ranging from the basics to open research problems and the latest advances in the field;

- Exercises at the end of each chapter that expand upon and support the material;

- Emphasis on the computational aspects of the theory; algorithms are presented to provide effective calculations;

- Many examples that illustrate the concepts and algorithms;

- Presentation of various connections between numerical semigroups and number theory, coding theory, algebraic geometry, linear programming, and commutative algebra would be of significant interest to researchers.

"Numerical Semigroups" is accessible to first year graduate students, with only a basic knowledge of algebra required, giving the full background needed for readers not familiar with the topic. Researchers will find the tools presented useful in producing examples and counterexamples in other fields such as algebraic geometry, number theory, and linear programming.

Keywords

  • Additive Semigroups
  • Embedding Dimension
  • Frobenius Number
  • Group theory
  • Irreducible
  • Modular
  • Monoid
  • Number theory
  • Numerical Semigroups
  • coding theory

Reviews

From the reviews:

“This book is a monograph, a summary of papers written on a … specialized subject. … In the body of the book, each topic … is accompanied by explanations as to which reference they appear in. … the writing is extremely clear, and the ideas well motivated. … The exercises at the end of the chapters are well taken … . without boring the reader or becoming too lengthy, the book does a very impressive job of being self-contained, and friendly to readers and students.” (Marion Cohen, The Mathematical Association of America, June, 2010)

“It is not often you can read a book in mathematics without a lot of background knowledge, and still reach the front of research of the subject. This is in fact possible with this book. … The exercises after all chapters help the reader to understand the concepts, and some of them also extend the subject by introducing new concepts. … all basic concepts are introduced in a purely elementary way, and to understand this book, one needs no background in algebraic geometry.”­­­ (Ralf Froberg, Semigroup Forum, Vol. 81, 2010)

“This book presents the theory of numerical semigroups, the development of which essentially is due to the two authors. They collected and proved all the results obtained so far in a very clear and self-contained way. Hence the text will be of interest for everyone working in the above mentioned fields and for everybody interested in the set of natural numbers, the understanding of which is heavily supported by the abstract theory of semigroups … .” (H. Mitsch, Monatshefte für Mathematik, Vol. 161 (1), August, 2010)

“This book gives a very readable account … that should be more widely known and will be useful to both researchers in the area and students new to the subject. … The exposition in the book is very good and can be followed even by talented undergraduates. … It is enjoyable and easy to read and serves as both a good way to learn a new subject for those who do not know much about the area, and a very useful research reference for experts.” (Nathan Kaplan, Zentralblatt MATH, Vol. 1220, 2011)

Authors and Affiliations

  • Faculty of Sciences, Department of Algebra, University of Granada, Granada, Spain

    J.C. Rosales, P. A. García-Sánchez

Bibliographic Information

Buying options

eBook USD 84.99
Price excludes VAT (USA)
  • ISBN: 978-1-4419-0160-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 109.99
Price excludes VAT (USA)
Hardcover Book USD 109.99
Price excludes VAT (USA)