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(s): Alfred Geroldinger, Franz Halter-Koch
Series: Pure and Applied Mathematics
Edition: 1
Publisher: Chapman and Hall CRC
Year: 2006
Language: English
Pages: 693
Non-Unique Factorizations Algebraic, Combinatorial & Analytic Theory......Page 1
Foreword......Page 6
Preface to this volume......Page 7
Acknowledgements......Page 10
Prerequisites......Page 11
Table of Contents......Page 17
1.1. Atoms & primes......Page 20
1.2. Free monoids, factorial monoids & factorizations......Page 29
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 & 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 & localizations......Page 72
2.3. Complete integral closures & Krull monoids......Page 78
2.4. Divisor homomorphisms & divisor theories......Page 86
2.5. Krull monoids & class groups......Page 94
2.6. Defining systems & v-noetherian monoids......Page 100
2.7. Finitary monoids......Page 111
2.8. Class semigroups......Page 122
2.9. C-monoids & finitely primary monoids......Page 131
2.10. Integral domains......Page 145
2.11. Congruence monoids & 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 & Krull monoids......Page 200
3.5. Type monoids......Page 215
3.6. Faithfully saturated submonoids......Page 221
a. Weakly Krull & one-dimensional domains......Page 228
b. K+M-domains......Page 235
c. Krull & Dedekind domains......Page 237
d. Mori domains & 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 & complete s-ideals in finitary monoids......Page 281
4.5. Products of strongly primary monoids & their submonoids......Page 293
4.6. C-monoids......Page 298
4.7. Integral domains & congruence monoids......Page 302
4.8. Realization theorems & 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 & p-groups......Page 350
5.6. Coverings by cosets & elementary p-groups......Page 359
5.7. Short zero-sum sequences & 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 & 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 & 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 & 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 & 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
