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

Let M be a commutative cancellative monoid. The set ∆(M), which consists of all positive integers which are distances between consecutive irreducible factorization lengths of elements in M , is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with divisor class group Zn, then it is well-known that ∆(M) ⊆ {1, 2, . . . , n − 2}. Moreover, equality holds for this containment when each divisor class of Zn contains a prime divisor from M . In this note, we consider the question of determining which subsets of {1, 2, . . . , n − 2} occur as the delta set of an individual element from M . We first prove for x ∈M that if n− 2 ∈ ∆(x), then ∆(x) = {n− 2} (i.e., not all subsets of {1, 2, . . . , 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 cyclic divisor class group: for every natural number m, there exist a natural number n and Krull monoid M with divisor class group Zn such that M has an element x with |∆(x)|≥ m.