Non-unique factorizations : algebraic, combinatorial and analytic theory

✍ Scribed by Alfred Geroldinger; Franz Halter-Koch

• Publisher: Taylor
• Year: 2006
• Tongue: English
• Leaves: 706
• Series: Monographs and textbooks in pure and applied mathematics, 278
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Non-Unique Factorizations Algebraic, Combinatorial and Analytic Theory......Page 4
PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes......Page 2
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS......Page 3
Foreword......Page 6
Preface to this volume......Page 7
Acknowledgements......Page 10
Prerequisites......Page 11
Table of Contents......Page 17
1.1. Atoms and primes......Page 20
1.2. Free monoids, factorial monoids and factorizations......Page 28
1.3. BF-monoids......Page 35
1.4. Systems of sets of lengths......Page 39
1.5. FF-monoids......Page 48
1.6. The catenary degree and the tame degree......Page 50
1.7. Rings of integers of algebraic number fields......Page 59
CHAPTER 2: Algebraic theory of monoids......Page 65
2.1. v-ideals......Page 66
2.2. Prime ideals and localizations......Page 72
2.3. Complete integral closures and Krull monoids......Page 78
2.4. Divisor homomorphisms and divisor theories......Page 86
2.5. Krull monoids and class groups......Page 94
2.6. Defining systems and v-noetherian monoids......Page 100
2.7. Finitary monoids......Page 111
2.8. Class semigroups......Page 122
2.9. C-monoids and finitely primary monoids......Page 131
2.10. Integral domains......Page 145
2.11. Congruence monoids and orders......Page 161
3.1. Finitary monoids......Page 179
3.2. Transfer principles......Page 187
3.3. C-monoids......Page 196
3.4. Saturated submonoids and Krull monoids......Page 200
3.5. Type monoids......Page 215
3.6. Faithfully saturated submonoids......Page 221
a. Weakly Krull and one-dimensional domains......Page 228
b. K+M-domains......Page 235
c. Krull and Dedekind domains......Page 237
d. Mori domains and congruence monoids......Page 240
e. Half-factorial quadratic orders......Page 244
3.8. Factorizations of powers of an element......Page 247
4.1. Multidimensional arithmetical progressions......Page 253
4.2. Almost arithmetical multiprogressions......Page 257
4.3. An abstract Structure Theorem for Sets of Lengths......Page 272
4.4. Pattern ideals and complete s-ideals in finitary monoids......Page 281
4.5. Products of strongly primary monoids and their submonoids......Page 293
4.6. C-monoids......Page 298
4.7. Integral domains and congruence monoids......Page 302
4.8. Realization theorems and further examples......Page 304
4.9. Sets of lengths of powers of an element......Page 313
5.1. Sequences over abelian groups......Page 319
5.2. Addition theorems......Page 333
5.3. Zero-sumfree sequences......Page 338
5.4. Cyclic groups......Page 344
5.5. Group algebras and p-groups......Page 350
5.6. Coverings by cosets and elementary p-groups......Page 359
5.7. Short zero-sum sequences and the inductive method......Page 365
5.8. Groups of rank two......Page 377
6.1. The generalized Davenport constants......Page 392
6.2. The Narkiewicz constants......Page 396
6.3. The elasticity and its refinements......Page 408
6.4. The catenary degree......Page 413
6.5. The tame degree......Page 418
6.6. Sets of lengths containing 2......Page 425
6.7. The set of distances and maximal half-factorial sets......Page 430
6.8. Minimal non-half-factorial sets......Page 444
7.1. Arithmetical characterizations of class groups I......Page 456
7.2. Arithmetical characterizations of class groups II......Page 465
7.3. The system of sets of lengths for finite abelian groups......Page 477
7.4. The system of sets of lengths for infinite abelian groups......Page 481
7.5. Additively closed sequences and restricted sumsets......Page 490
7.6. Factorization of large elements......Page 499
CHAPTER 8: Abstract analytic number theory......Page 520
8.1. Dirichlet series......Page 521
8.2. A general Tauberian theorem......Page 531
8.3. Abstract formations and zeta functions......Page 544
8.4. Arithmetical formations I: Zeta functions......Page 552
8.5. Arithmetical formations II: Asymptotic results......Page 563
8.6. Arithmetical formations III: Structure theory......Page 577
8.7. Geometrical formations I: Asymptotic results......Page 587
8.8. Geometrical formations II: Structure theory......Page 602
8.9. Algebraic function fields......Page 605
8.10. Obstructed formations......Page 616
CHAPTER 9: Analytic theory of non-unique factorizations......Page 627
9.1. Analytic theory of types......Page 628
9.2. Elements with prescribed factorization properties......Page 644
9.3. The number of distinct factorizations......Page 649
9.4. Block-dependent factorization properties......Page 652
APPENDIX A: Abelian Groups......Page 664
APPENDIX B: Complex Analysis......Page 673
APPENDIX C: Theory of Integration......Page 684
APPENDIX D: Polyhedral Cones......Page 689
Bibliography......Page 692
List of Symbols......Page 705