The Monotone Catenary Degree of Krull Monoids

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.