Let N be the set of nonnegative integers. A numerical semigroup is a nonempty subset S of N that is closed under addition, contains the zero element, and whose complement in N is ?nite. If n ,...,n are positive integers with gcd{n ,...,n } = 1, then the set hn ,..., 1 e 1 e 1 n i = {? n +··· + ? n | ? ,...,? ? N} is a numerical semigroup. Every numer e 1 1 e e 1 e ical semigroup is of this form. The simplicity of this concept makes it possible to state problems that are easy to understand but whose resolution is far from being trivial. This fact attracted several mathematicians like Frobenius and Sylvester at the end of the 19th century. This is how for instance the Frobenius problem arose, concerned with ?nding a formula depending on n ,...,n for the largest integer not belonging to hn ,...,n i (see [52] 1 e 1 e for a nice state of the art on this problem).

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.

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 goodway 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)