Combinatorial Non-Unique Factorization Theory

Each non-zero and non-invertible element of the ring of integers of an algebraic number field is the product of finitely many irreducible elements, yet in general an element has various essentially distinct factorizations. Moreover, it is well-known that the ring of integers of an algebraic number field is a unique factorization domain if and only if its class group is trivial. A classical aim of Non-Unique Factorization Theory is to understand the various phenomena of non-uniqueness that arise in this context. Yet, other domains and monoids for which it makes sense to investigated factorization are considered as well. The following methodology has turned out to be useful. Investigate some class of domains or monoids from an algebraic point of view, in particular investigate its ideals. Based on the thus gained knowledge construct auxiliary monoids and identify a way to transfer the problems of Non-Unique Factorization Theory in these domains or monoids to the auxiliary monoids. Then, investigate the problems in the auxiliary monoids and transfer the obtained results back to the original domains or monoids. The former problems are called algebraic and the latter ones combinatorial. Problems that are, in the sense above, combinatorial will be considered. More specifically, for rings of algebraic integers and more generally Krull monoids it is well-known how to transfer (most) problems on factorizations to auxiliary monoids. The arising problems are problems on zero-sum sequences over subsets of the class groups. The focus will be on problems related to the structure of sets of lengths of elements of Krull monoids with finite class group; a non-negative integer n is contained in the set of lengths of an element if this element is the product of n irreducible elements. It is well-known that in this case sets of lengths have some structure, namely up to some quantifiable exceptions they consist of a certain set, called period, that is repeated, i.e., they are generalized arithmetical progressions. These periods will be investigated, and closely related to this problem subsets of the class group yielding sets of lengths with a prescribed minimal distance or trivial ones, i.e., half-factorial sets, will be considered. Among others, results on sum-free subsets of finite abelian groups and related results and methods will be applied.