The Catenary Degree of Krull Monoids I

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each two factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_0,...,z_k=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf{c}(H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf{c}(H)$ and characterize when $\mathsf{c}(H)\le 4$.

References in corpus (1)

• The catenary degree of Krull monoids II

Cited by in corpus (11)

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• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
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• The catenary degree of Krull monoids II
• A Characterization of class groups via sets of lengths {II}
• Long sets of lengths with maximal elasticity
• Sets of minimal distances and characterizations of class groups of Krull monoids
• Minimal relations and catenary degrees in Krull monoids
• The set of distances in seminormal weakly Krull monoids
• A characterization of finite abelian groups via sets of lengths in transfer Krull monoids