Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids

TL;DR

This paper investigates the structure of sets of lengths in transfer Krull monoids and weakly Krull monoids, providing new descriptions for groups with Davenport constant 5 and highlighting differences between these monoid classes.

Contribution

It offers a complete description of the system of sets of lengths for transfer Krull monoids over groups with Davenport constant 5 and compares these systems with those of weakly Krull monoids.

Findings

Complete description for groups with D(G)=5

Sets of lengths satisfy the Structure Theorem under finiteness assumptions

Systems of lengths differ between transfer Krull and weakly Krull monoids

Abstract

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.