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 cmon (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 = z0, ... ,zk = z′ of a with increasing lengths—that is, |z0| ≤  ... ≤  |zk|—such that, for each $${i \in [1,k]}$$ , zi arises from zi-1 by replacing at most m atoms from zi-1 by at most m new atoms. Up to now there was only an abstract finiteness result for cmon (H), but the present paper offers the first explicit upper and lower bounds for cmon (H) in terms of the group invariants of G.

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