Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory

by Geroldinger; Alfred

• ISBN13: 9781584885764
• ISBN10: 1584885769
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-01-13
• Publisher: Chapman & Hall/

Summary

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.

Author Biography

Franz Halter-Koch is a Professor of Mathematics at Karl-Franzens-Universitat Graz, Austria.

Table of Contents

Concepts in factorization theory and examples	p. 1
Atoms and primes	p. 1
Free monoids, factorial monoids and factorizations	p. 9
BF-monoids	p. 16
Systems of sets of lengths	p. 20
FF-monoids	p. 29
The catenary degree and the tame degree	p. 31
Rings of integers of algebraic number fields	p. 40
Algebraic theory of monoids	p. 47
v-ideals	p. 48
Prime ideals and localizations	p. 54
Complete integral closures and Krull monoids	p. 60
Divisor homomorphisms and divisor theories	p. 68
Krull monoids and class groups	p. 76
Defining systems and v-noetherian monoids	p. 82
Finitary monoids	p. 93
Class semigroups	p. 104
C-monoids and finitely primary monoids	p. 113
Integral domains	p. 127
Congruence monoids and orders	p. 143
Arithmetic theory of monoids	p. 161
Finitary monoids	p. 161
Transfer principles	p. 169
C-monoids	p. 178
Saturated submonoids and Krull monoids	p. 182
Type monoids	p. 197
Faithfully saturated submonoids	p. 203
Integral domains and congruence monoids	p. 210
Factorizations of powers of an element	p. 229
The structure of sets of lengths	p. 235
Multidimensional arithmetical progressions	p. 235
Almost arithmetical multiprogressions	p. 239
An abstract Structure Theorem for Sets of Lengths	p. 254
Pattern ideals and complete s-ideals in finitary monoids	p. 263
Products of strongly primary monoids and their submonoids	p. 275
C-monoids	p. 280
Integral domains and congruence monoids	p. 284
Realization theorems and further examples	p. 286
Sets of lengths of powers of an element	p. 295
Additive group theory	p. 301
Sequences over abelian groups	p. 301
Addition theorems	p. 315
Zero-sumfree sequences	p. 320
Cyclic groups	p. 326
Group algebras and p-groups	p. 332
Coverings by cosets and elementary p-groups	p. 341
Short zero-sum sequences and the inductive method	p. 347
Groups of rank two	p. 359
Arithmetical invariants of Krull monoids	p. 375
The generalized Davenport constants	p. 375
The Narkiewicz constants	p. 379
The elasticity and its refinements	p. 391
The catenary degree	p. 396
The tame degree	p. 401
Sets of lengths containing 2	p. 408
The set of distances and maximal half-factorial sets	p. 413
Minimal non-half-factorial sets	p. 427
Global arithmetic of Krull monoids	p. 439
Arithmetical characterizations of class groups I	p. 439
Arithmetical characterizations of class groups II	p. 448
The system of sets of lengths for finite abelian groups	p. 460
The system of sets of lengths for infinite abelian groups	p. 464
Additively closed sequences and restricted sumsets	p. 473
Factorization of large elements	p. 482
Abstract analytic number theory	p. 503
Dirichlet series	p. 504
A general Tauberian theorem	p. 514
Abstract formations and zeta functions	p. 527
Arithmetical formations I: Zeta functions	p. 535
Arithmetical formations II: Asymptotic results	p. 546
Arithmetical formations III: Structure theory	p. 560
Geometrical formations I: Asymptotic results	p. 570
Geometrical formations II: Structure theory	p. 585
Algebraic function fields	p. 588
Obstructed formations	p. 599
Analytic theory of non-unique factorizations	p. 611
Analytic theory of types	p. 612
Elements with prescribed factorization properties	p. 628
The number of distinct factorizations	p. 633
Block-dependent factorization properties	p. 636
Abelian Groups	p. 649
Complex Analysis	p. 659
Theory of Integration	p. 671
Polyhedral Cones	p. 677
Bibliography	p. 681
List of Symbols	p. 695
Subject Index	p. 697
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