Additive Group Theory and Non-Unique Factorizations

The theory of non-unique factorizations has its origin in the theory of algebraic numbers. An integral domain is called factorial or a unique factorization domain if every non-zero non-unit has a factorization into (finitely many) irreducible elements (atoms) and the factorization is essentially unique (that is, unique up to associates and the order of the factors). The ring of integers of an algebraic number field has, like every noetherian domain, the property that every non-zero non-unit is a product of irreducible elements (atoms), but in general there are many essentially distinct factorizations. It is the main objective of factorization theory to describe and to classify the various phenomena of non-uniqueness of factorizations in an integral domain R in terms of algebraic invariants of R. If R is a Krull domain, then most phenomena only depend on the class group of R and on the distribution of the prime divisors in the classes. Thus they can be studied via the monoid of zero-sum sequences over the class group. The connection is most close if the class group is finite and every class contains a prime. This is the case for rings of integers of algebraic number fields. Additive group theory has its origin in additive number theory. Starting from classical addition theorems this field saw a rapid development initiated by the work of P. Erdös, G.A. Freiman, M. Kneser and H.B. Mann. The classical topic is the investigation of (the structure of) the sumset A+B = {a+b | a lies in A and b lies in B} where A and B are (finite) subsets of some given abelian group G. Conversely, starting from a sumset, which satisfies some extremal condition, one tries to get information on the possible structure of the summands (keywords: direct and inverse additive problems). Inspired by questions from combinatorial number theory the above problems have their canonical analogues for sequences over the abelian group G (or in other words, for multisets). Both fields, factorization theory (in Krull monoids) and additive group theory, are closely connected (via the monoid of zero-sum sequences over the class group of the Krull monoid). Some central invariants in factorization theory, such as the Davenport constant, have their independent tradition and significance in additive group theory. What is even more important is that central methods in factorization theory (over Krull monoids) are based on results from additive group theory (keywords: addition theorems). The following problems are in the center of interest for the project under application: Combinatorial invariants of the theory of non-unique factorizations: We concentrate on the (generalized) Davenport constants and the cross number of finite abelian groups. These invariants occur in a natural way in factorization theory, both in a direct way as well as control parameters for other invariants from factorization theory (as the catenary degree and the set of distances). Most of the results so far are restricted to p-groups and to groups of rank at most two. Inverse (zero-sum) problems over finite abelian groups G: The investigation of inverse problems connected to the classical invariants D(G), s(G) and Eta (G) goes back to the 1960s but until very recently most work is restricted to the cyclic case. First results for groups of rank two are based on strong addition theorems. Answers to these inverse problems give among others information on the structure of short sets of lengths.