SETS OF CROSS NUMBERS OF SEQUENCES OVER FINITE ABELIAN GROUPS
Résumé
Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial zero-sum free sequences over $G$.
It is clear that $\mathsf W(G)$ and $\mathsf w(G)$ are bounded subsets of $\frac{1}{\exp(G)}\mathbb{N}$ with maximum $ \mathsf K(G)$ and $\mathsf k(G)$, respectively (here $\mathsf{K}(G)$ and $\mathsf{k}(G)$ denote the large and the small cross number of $G$, respectively).
We give results on the structure of $\mathsf W(G)$ and $\mathsf w(G)$.
We first show that both sets contain long arithmetic progressions and that only close to the maximum there might be some gaps.
Then, we provide groups for which $\mathsf W(G)$ and $\mathsf w(G)$ actually are arithmetic progressions, and argue that this is rather a rare phenomenon.
Finally, we provide some results in case there are gaps.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)