Numerical Semigroups (Developments in Mathematics Book 20)

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. Notable Elements -- Numerical Semigroups With Maximal Embedding Dimension -- Irreducible Numerical Semigroups -- Proportionally Modular Numerical Semigroups -- The Quotient Of A Numerical Semigroup By A Positive Integer -- Families Of Numerical Semigroups Closed Under Finite Intersections And Adjoin Of The Frobenius Number -- Presentations Of A Numerical Semigroup -- The Gluing Of Numerical Semigroups -- Numerical Semigroups With Embedding Dimension Three -- The Structure Of A Numerical Semigroup. By J.c. Rosales, P.a. García-sánchez. Includes Bibliographical References (p. 171-175) And Index. 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). Front Matter....Pages i-ix Introduction....Pages 1-3 Notable elements....Pages 5-18 Numerical semigroups with maximal embedding dimension....Pages 19-32 Irreducible numerical semigroups....Pages 33-55 Proportionally modular numerical semigroups....Pages 57-76 The quotient of a numerical semigroup by a positive integer....Pages 77-90 Families of numerical semigroups closed under finite intersections and adjoin of the Frobenius number....Pages 91-104 Presentations of a numerical semigroup....Pages 105-122 The gluing of numerical semigroups....Pages 123-136 Numerical semigroups with embedding dimension three....Pages 137-154 The structure of a numerical semigroup....Pages 155-169 Back Matter....Pages 171-181 "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." ""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."--Jacket "Numerical Semigroups" is the first monograph devoted exclusively to the development of the theory of numerical semigroups. This concise, self-contained text is accessible to first year graduate students, giving the full background needed for readers unfamiliar 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.