\(\omega \)-primality in arithmetic Leamer monoids
The paper under review studies \(\omega\)-primality for Leamer monoids. For an additive monoid \(S\) and some \(x \in S\), not a unit, one defines \(\omega(x)\) as the smallest \(m\) such that whenever \(x\) divides \(x_1 + \dots +x_k\) with \(x_i\in S\) there is a subset \(I\) of \(\{1, \dots, k\}\) of at most \(m\) elements such that \(x\) divides \(\sum_{i\in I}x_i\). If \(x\) is a prime element, then \(\omega(x)=1\). In a way the \(\omega\)-function allows to quantify how far an element is from being prime. A numerical semigroup \(\Gamma\) is a submonoid of \((\mathbb{N}, +)\) with finite complement. For \(s \in \mathbb{N} \setminus \Gamma\) one defines \(S_{\Gamma}^s = \{(0,0)\} \cup \{(x,n) \colon \{x, x+s, x+2s, \dots, x+ns\} \subset \Gamma\} \subset \mathbb{N}^2\). This is a monoid called a Leamer monoid. The authors study the \(\omega\)-function for Leamer monoids, and more specifically Leamer monoids for \(\Gamma\) being generated by an arithmetical progression. In this case exact results are obtained.
scientific article; zbMATH DE number 7080903
scientific article