On the delta set and the Betti elements of a BF-monoid.
A monoid \(S\) is called a bounded factorization monoid, or BF-monoid, if for any \(x\in S\) the set of lengths of possible factorizations of \(S\) is bounded. In this paper the authors investigate certain factorization invariants of BF-monoids, focusing on the Delta set of \(S\), denoted \(\Delta(S)\). Let \(x\in S\) and \(\mathcal L(x)=\{m_1<m_2<\cdots<m_k\}\) denote the set of lengths of the factorizations of \(x\). Then \(\Delta(x)=\{m_i-m_{i-1}\mid 2\leq i\leq k\}\). The Delta set of \(S\) is the union of the Delta sets of the elements of \(S\). The Delta set of a BF-monoid \(S\) is one measure of how far \(S\) is from being a unique factorization domain. For example, \(\Delta(S)\) is empty if and only if for each \(x\in S\) all factorizations of \(x\) have the same length, and \(\Delta(S)=\{k\}\) if and only if for each \(x\in S\) the set of lengths of \(x\) is an arithmetic progression of difference \(k\). Factorization properties of BF-monoids can be quite subtle and even monoids that are easy to describe can have complicated Delta sets. The authors show that for a finitely presented BF-monoid \(S\) the minimum and maximum of \(\Delta(S)\) can be determined by focusing on the factorizations of a certain finite subset of the elements of \(S\) known as the Betti elements of \(S\). In the particular case where \(S\) is a numerical monoid with a minimal generating set of size \(3\) they give an explicit description of \(\max\Delta(S)\) in terms of the generators and use it to give a description of when \(\Delta(S)\) consists of a single element.
scientific article; zbMATH DE number 6116564
scientific article