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

TL;DR

This paper investigates which subsets of possible delta sets occur for elements in Krull monoids with cyclic class groups, revealing restrictions on realizable delta sets and demonstrating the unbounded complexity of these sets.

Contribution

It characterizes the delta sets of individual elements in Krull monoids with cyclic class groups, showing certain subsets cannot occur and establishing an unbounded diversity of delta sets.

Findings

If n-2 is in the delta set, then the delta set is exactly {n-2}

Not all subsets of {1,..., n-2} are realizable as delta sets of individual elements

For any natural number m, there exists a Krull monoid with an element whose delta set has size at least m

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 $|\Delta(x)| \ge m$.