Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group

Abstract

Some of the fundamental notions related to sets of lengths of Krull monoids with finite class group are discussed, and a survey of recent results is given. These include the elasticity and related notions, the set of distances, and the structure theorem for sets of lengths. Several open problems are mentioned.

The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics

“…There is a variety of transfer homomorphisms from monoids of zero-sum sequences to monoids of zero-sum sequences in order to simplify specific structural features of the involved subsets of groups. Below we present a simple example of such a transfer homomorphism which we will meet again in Proposition 4.9 (for more of this nature we refer to [74] and to [40,Theorem 6.7.11]). Let G be abelian and let G 0 ⊂ G be a subset.…”

Definite orders with locally free cancellation

“…The fact that is a transfer Krull monoid of finite type, implies that many questions on factorizations in , in particular all the ones on sets of lengths, can be reduced to questions in combinatorial and additive number theory over finite abelian groups, specifically the stable class group . See the surveys as a starting point into the extensive literature; and for recent progress. In particular, the set of distances is finite, indeed for some .…”

On the algebraic and arithmetic structure of the monoid of product-one sequences II

“…S · S −1 = a [4] · (a 3 ) [2] · b [2] · (a 3 ) [2] · (a 2 b) [2] , whence {2, 3, 6} ⊂ L(S · S −1 ). On the other hand, by [32,Proposition 4.14], there is no L ∈ L(C 2 2 ⊕ C 4 ) such that {2, 3, 6} ⊂ L. Thus L(C 2 2 ⊕ C 4 ) = L(D 8 ).…”