Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory

ISBN-13: 9781584885764 / Angielski / Twarda / 2006 / 728 str.

Alfred Geroldinger; Franz Halter-Koch

Opis

From its origins in algebraic number theory, the theory of non-unique factorizations has emerged as an independent branch of algebra and number theory. Focused efforts over the past few decades have wrought a great number and variety of results. However, these remain dispersed throughout the vast literature. For the first time, Non-Unique Factorizations: Algebraic, Combinatorial, and Analytic Theory offers a look at the present state of the theory in a single, unified resource.
Taking a broad look at the algebraic, combinatorial, and analytic fundamentals, this book derives factorization results and applies them in concrete arithmetical situations using appropriate transfer principles. It begins with a basic introduction that can be understood with knowledge of standard basic algebra. The authors then move to the algebraic theory of monoids, arithmetic theory of monoids, the structure of sets of lengths, additive group theory, arithmetical invariants, and the arithmetic of Krull monoids. They also provide a self-contained introduction to abstract analytic number theory as well as a modern treatment of W. Narkiewicz's analytic theory of non-unique factorizations.
Non-Unique Factorizations: Algebraic, Combinatorial, and Analytic Theory builds the discussion from first principles to applied problem solving, making it ideally suited to those not familiar with the theory as well as those who wish to deepen their understanding.

Spis treści

CONCEPTS IN FACTORIZATION THEORY AND EXAMPLES
Atoms and Primes
Free Monoids, Factorial Monoids and Factorizations
BF-Monoids
Systems of Sets of Lengths
FF-Monoids
The Catenary Degree and the Tame Degree
Rings of Integers of Algebraic Number Fields
ALGEBRAIC THEORY OF MONOIDS
v-Ideals
Prime Ideals and Localizations
Complete Integral Closures and Krull Monoids
Divisor Homomorphisms and Divisor Theories
Krull Monoids and Class Groups
Defining Systems and v-Noetherian Monoids
Finitary Monoids
Class Semigroups
C-Monoids and Finitely Primary Monoids
Integral Domains
Congruence Monoids and Orders
ARITHMETIC THEORY OF MONOIDS
Finitary Monoids
Transfer Principles
C-Monoids
Saturated Submonoids and Krull Monoids
Type Monoids
Faithfully Saturated Submonoids
Integral Domains and Congruence Monoids
Factorizations of Powers of an Element
THE STRUCTURE OF SETS OF LENGTHS
Multidimensional Arithmetical Progressions
Almost Arithmetical Multiprogressions
An Abstract Structure Theorem for Sets of Lengths
Pattern Ideals and Complete s-Ideals in Finitary Monoids
Products of Strongly Primary Monoids and their Submonoids
C-Monoids
Integral Domains and Congruence Monoids
Realization Theorems and Further Examples
Sets of Lengths of Powers of an Element
ADDITIVE GROUP THEORY
Sequences over Abelian Groups
Addition Theorems
Zero-Sumfree Sequences
Cyclic Groups
Group Algebras and p-Groups
Coverings by Cosets and Elementary p-Groups
Short Zero-Sum Sequences and the Inductive Method
Groups of Rank Two
ARITHMETICAL INVARIANTS OF KRULL MONOIDS
The Generalized Davenport Constants
The Narkiewicz Constants
The Elasticity and Its Refinements
The Catenary Degree
The Tame Degree
Sets of Lengths Containing 2
The Set of Distances and Maximal Half-Factorial Sets
Minimal Non-Half-Factorial Sets
GLOBAL ARITHMETIC OF KRULL MONOIDS
Arithmetical Characterizations of Class Groups I
Arithmetical Characterizations of Class Groups II
The System of Sets of Lengths for Finite Abelian Groups
The System of Sets of Lengths for Infinite Abelian Groups
Additively Closed Sequences and Restricted Sumsets
Factorization of Large Elements
ABSTRACT ANALYTIC NUMBER THEORY
Dirichlet Series
A General Tauberian Theorem
Abstract Formations and Zeta Functions
Arithmetical Formations I: Zeta Functions
Arithmetical Formations II: Asymptotic Results
Arithmetical Formations III: Structure Theory
Geometrical Formations I: Asymptotic Results
Geometrical Formations II: Structure Theory
Algebraic Function Fields
Obstructed Formations
ANALYTIC THEORY OF NON-UNIQUE FACTORIZATIONS
Analytic Theory of Types
Elements with Prescribed Factorization Properties
The Number of Distinct Factorizations
Block-Dependent Factorization Properties
APPENDIX A: ABELIAN GROUPS
APPENDIX B: COMPLEX ANALYSIS
APPENDIX C: THEORY OF INTEGRATION
APPENDIX D: POLYHEDRAL CONES
BIBLIOGRAPHY
LIST OF SYMBOLS
SUBJECT INDEX

O autorze

Alfred Geroldinger, Franz Halter-Koch