1st Edition

Non-Unique Factorizations Algebraic, Combinatorial and Analytic Theory

    728 Pages 7 B/W Illustrations
    by Chapman & Hall

    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.

    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

    Biography

    Alfred Geroldinger, Franz Halter-Koch

    "Combining methods of various branches of mathematics, it brings together a theory from classical results to topics reflecting the recent ideas. It is a nice book written in a precise, readable style."

    – In EMS Newsletter, September 2007