Combinatorial Factorization Theory

This monograph invites readers into the study of factorization theory of Krull monoids and domains. A first of its kind, the book bridges the algebraic background and the specialized machinery of additive combinatorics needed for researchers interested in factorization theory. Beginning with the necessary tools from algebra, the opening chapter focuses on the arithmetic of monoids and its link with additive combinatorics. From here, chapters discuss the methods and results from relevant areas of additive combinatorics, with a special focus on zero-sum problems. In the latter chapters, these tools are used to study factorization in monoids of zero-sum sequences over finite abelian groups. In particular, length sets and the study of elasticities, along with the characterization problem for systems of length sets, are a highlight of the volume. All of these results then apply to factorization in any Krull monoid or Krull domain. With its careful curation of the last 20 years of research, Combinatorial Factorization Theory serves as a guiding research monograph for this growing area. The introductory materials make this book an appropriate entry point for experienced and prospective researchers interested in learning more about these fields. A grounding in graduate algebra is assumed, but no specialist knowledge is required.