Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids

  • Alfred Geroldinger
  • Wolfgang A. Schmid
  • Qinghai Zhong


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.


Transfer Krull monoids Weakly Krull monoids Sets of lengths Zero-sum sequences 



This work was supported by the Austrian Science Fund FWF, Project Number P28864-N35, and the ANR project Caesar, project number ANR-12-BS01-0011


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alfred Geroldinger
    • 1
  • Wolfgang A. Schmid
    • 2
    • 3
  • Qinghai Zhong
    • 1
  1. 1.Institute for MathematicsUniversity of GrazGrazAustria
  2. 2.Laboratoire Analyse, Géométrie et Applications (LAGA, UMR 7539, CNRS)COMUE Université Paris LumièresSaint-Denis cedexFrance
  3. 3.Université Paris 13Sorbonne Paris CitéVilletaneuseFrance

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