On delta sets and their realizable subsets in Krull monoids with cyclic class groups. (Q5891844)

Let \(M\) be a commutative cancellative monoid. The paper under review is concerned with the set \(\Delta(M)\), which consists of all positive integers which are distances between consecutive factorization lengths of elements in \(M\). This invariant is a widely studied object in the theory of nonunique factorizations. For instance, when \(M\) is a Krull monoid with cyclic class group of order \(n\geq 3\), by Theorem 6.7.1 of \textit{A. Geroldinger} and \textit{F. Halter-Koch} [Non-unique factorizations. Algebraic, combinatorial and analytic theory. Pure and Applied Mathematics (Boca Raton) 278. Boca Raton: Chapman \& Hall/CRC (2006; Zbl 1113.11002)] it is known that \(\Delta(M)=\{1,\ldots,n-2\}\). Related to this equality, the authors exploit the question: Which subsets \(T\subset\{1,\ldots,n-2\}\) are realized as the delta set of an individual element in \(M\) (i.e., for which \(T\) does there exist an \(x\in M\) such that \(T=\Delta(x)\))? They show that when \(x\in M\) with \(n-2\in\Delta(x)\), then \(\Delta(x)\) is exactly \(\{n-2\}\). They also show that for every natural number \(m\), there exists a Krull monoid \(M\) with finite class group such that \(M\) has an element \(x\) with \(|\Delta(x)|\geq m\).