ON THE INCOMPARABILITY OF SYSTEMS OF SETS OF LENGTHS

Alfred Geroldinger (1) , Wolfgang Schmid (2)

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

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $\mathcal L (H)$ of all sets of lengths of $H$ and study when $\mathcal L (H)$ contains or is contained in a system $\mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $\mathsf D (G')=\mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m \ge 7$ or an elementary $2$-group of rank $m-1 \ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $\mathsf D (G')=\mathsf D (G)$, then the systems $\mathcal L (H)$ and $\mathcal L (H')$ are incomparable.

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

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

Alfred Geroldinger, Wolfgang Schmid. ON THE INCOMPARABILITY OF SYSTEMS OF SETS OF LENGTHS. 2020. ⟨hal-02566257v2⟩