On the incomparability of systems of sets of lengths

2020/05/07 by Geroldinger, Alfred, Schmid, Wolfgang
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2005.03316

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. We consider the system \mathcal L (H) of all sets of lengths of H and study when \mathcal L (H) contains or is contained in a system \mathcal L (H') of a Krull monoid H' with finite class group G', prime divisors in all classes and Davenport constant \mathsf D (G')=\mathsf D (G). Among others, we show that if G is either cyclic of order m ≥ 7 or an elementary 2-group of rank m-1 ≥ 6, and G' is any group which is non-isomorphic to G but with Davenport constant \mathsf D (G')=\mathsf D (G), then the systems \mathcal L (H) and \mathcal L (H') are incomparable.

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