On Delta Sets and their Realizable Subsets in Krull Monoids with Cyclic Class Groups

Abstract

Let $M$ be a commutative cancellative monoid. The set $\Delta (M)$, which consists of all positive integers which are distances between consecutive factorization lengths of elements in $M$, is a widely studied object in the theory of nonunique factorizations. If $M$ is a Krull monoid with cyclic class group of order $n\ge 3$, then it is well-known that $\Delta (M)\subseteq \{1,\dots ,n-2\}$. Moreover, equality holds for this containment when each class contains a prime divisor from $M$. In this note, we consider the question of determining which subsets of $\{1,\dots ,n-2\}$ occur as the delta set of an individual element from $M$. We first prove for $x\in M$ that if $n-2\in \Delta (x)$, then $\Delta (x)=\{n-2\}$ (i.e., not all subsets of $\{1,\dots ,n-2\}$ can be realized as delta sets of individual elements). We close by proving an Archimedean-type property for delta sets from Krull monoids with finite cyclic class group: for every natural number m, there exist a Krull monoid $M$ with finite cyclic class group such that $M$ has an element $x$ with $|\Delta (x)|\ge m$.

10 pages