Short description:

This monograph-the first of its kind-presents the current state of knowledge on the theory of non-unique factorizations along with a broad discussion of its algebraic, combinatorial, and analytic fundamentals. The book focuses on factorization properties in rings of integers, orders of algebraic number fields, holomorphy rings, Krull and Mori domains, Krull monoids, congruence monoids, Hilbert monoids, and zero-sum sequences over Abelian groups. It also includes self-contained introductions to the v-ideal theory of monoids, additive group theory, and abstract analytic number theory. Within this treatment, the authors unify, extend, and put into a new context many known algebraic, combinatorial, and analytic results.

Long description:

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.

"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

Table of Contents:

Concepts in Factorization Theory and Examples. Algebraic Theory of Monoids. Arithmetic Theory of Monoids. The structure of Sets of Lengths. Additive Group Theory. Arithmetical Invariants of Krull Monoids. Global Arithmetic of Krull Monoids. Abstract Analytic Number Theory. Analytic Theory of Non-Unique Factorizations. Appendix A: Abelian Groups. Appendix B: Complex Analysis. Appendix C: Theory of Integration. Appendix D: Polyhedral Cones. Bibliography. List of Symbols. Subject Index.