The Monotone Catenary Degree of Krull Monoids

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c mon (H) of H is the smallest integer m with the following property: for each $${a \in H}$$ and each two factorizations z, z′ of a with length |z| ≤ |z′|, there exist factorizations z = z 0, ... ,z k = z′ of a with increasing lengths—that is, |z 0| ≤ ... ≤ |z k |—such that, for each $${i \in [1,k]}$$ , z i arises from z i-1 by replacing at most m atoms from z i-1 by at most m new atoms. Up to now there was only an abstract finiteness result for c mon (H), but the present paper offers the first explicit upper and lower bounds for c mon (H) in terms of the group invariants of G.