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2016 | Buch

Numerical Semigroups, the Basics Kapitel lesen Erstes Kapitel lesen

Numerical Semigroups and Applications

verfasst von: Abdallah Assi, Pedro A. García-Sánchez

Verlag: Springer International Publishing

Buchreihe : RSME Springer Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This work presents applications of numerical semigroups in Algebraic Geometry, Number Theory, and Coding Theory. Background on numerical semigroups is presented in the first two chapters, which introduce basic notation and fundamental concepts and irreducible numerical semigroups. The focus is in particular on free semigroups, which are irreducible; semigroups associated with planar curves are of this kind. The authors also introduce semigroups associated with irreducible meromorphic series, and show how these are used in order to present the properties of planar curves. Invariants of non-unique factorizations for numerical semigroups are also studied. These invariants are computationally accessible in this setting, and thus this monograph can be used as an introduction to Factorization Theory. Since factorizations and divisibility are strongly connected, the authors show some applications to AG Codes in the final section. The book will be of value for undergraduate students (especially those at a higher level) and also for researchers wishing to focus on the state of art in numerical semigroups research.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Numerical Semigroups, the Basics
Abstract
In this chapter we introduce the basic notions related to numerical semigroups. Numerical semigroups have not been always been referred to as such. In the past some authors called them semimodules, or demimodules and recently many authors (mainly those concerned with factorization properties) are starting to refer to them as numerical monoids.
Abdallah Assi, Pedro A. García-Sánchez
Chapter 2. Irreducible Numerical Semigroups
Abstract
A numerical semigroup S is irreducible it cannot be expressed as the intersection of two proper oversemigroups. The motivation of the study of these semigroups was initially to express any numerical semigroup as a finite intersection of irreducible numerical semigroups, and then derive properties of the original semigroup in terms of the irreducibles that appear in this decomposition. Historically this was not the reason to study these semigroups. It turns out that irreducible numerical semigroups are either symmetric (when their Frobenius number is odd) or pseudo-symmetric (even Frobenius number); and every symmetric or pseudo-symmetric numerical semigroup is irreducible.
Abdallah Assi, Pedro A. García-Sánchez
Chapter 3. Semigroup of an Irreducible Meromorphic Series
Abstract
Let \({\mathbb K}\) be an algebraically closed field of characteristic zero and let \(f(x,y)=y^n+a_1(x)y^{n-1}+\dots +a_n(x)\) be a nonzero polynomial of \({\mathbb K}(\!(x)\!)[y]\) where \({\mathbb K}(\!(x)\!)\) denotes the field of meromorphic series in x.
Abdallah Assi, Pedro A. García-Sánchez
Chapter 4. Minimal Presentations
Abstract
It is usual in Mathematics to represent objects by means of a free object modulo certain relations fulfilled by the generators of the free object. The reader familiar to group theory surely has used many times definitions of groups by means of generators and relations. Relations are usually represented as equalities, or simply words in the free group on the generators (this means that they are equal to the identity element; this is due to the fact that we have inverse in groups). Here we represent relations by pairs. These are pairs of factorizations of certain elements in the semigroup; they will be a crucial tool for studying factorizations in the next chapter.
Abdallah Assi, Pedro A. García-Sánchez
Chapter 5. Factorizations and Divisibility
Abstract
Let S be a numerical semigroup minimally generated by \(\{n_1,\ldots ,n_p\}\).
Abdallah Assi, Pedro A. García-Sánchez
Backmatter
Metadaten
Titel
Numerical Semigroups and Applications
verfasst von
Abdallah Assi
Pedro A. García-Sánchez
Copyright-Jahr
2016
Electronic ISBN
978-3-319-41330-3
Print ISBN
978-3-319-41329-7
DOI
https://doi.org/10.1007/978-3-319-41330-3

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