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The Monotone Catenary Degree of Krull Monoids
Authors:Alfred Geroldinger  Pingzhi Yuan
Institution:1. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstrasse 36, 8010, Graz, Austria
2. School of Mathematics, South China Normal University, 510631, Guangzhou, People’s Republic of China
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.
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