SYSTEMS OF SETS OF LENGTHS: TRANSFER KRULL MONOIDS VERSUS WEAKLY KRULL MONOIDS

Alfred Geroldinger (1) , Wolfgang Schmid (2) , Qinghai Zhong (1)

1 IM - Institut für Mathematik [Graz]
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.

Mots clés: transfer Krull monoids; weakly Krull monoids; sets of lengths; zero-sum sequences

Domaines: Algèbre commutative [math.AC]; Théorie des nombres [math.NT]; Combinatoire [math.CO]

Alfred Geroldinger, Wolfgang Schmid, Qinghai Zhong. SYSTEMS OF SETS OF LENGTHS: TRANSFER KRULL MONOIDS VERSUS WEAKLY KRULL MONOIDS. 2016. ⟨hal-01332417v1⟩