The genus of a quotient of several types of numerical semigroups (Q6594431)

Let \(\mathbb{N}_0\) be the set of non-negative integers. For a set \(A\subseteq \mathbb{N}_0\), denote \[\langle A\rangle=\{\lambda_1 x_1 + \cdots +\lambda_n x_n\mid n\in \mathbb{N}_0\setminus \{0\}, \lambda_1,\ldots,\lambda_n \in \mathbb{N}_0,a_1,\ldots,a_n\in A\}\] that is, the additive submonoid of \(\mathbb{N}_0\) generated by \(A\). Let \(S=\langle A\rangle\), it is known that \(\mathbb{N}_0\setminus S\) is a finite set if and only if \(\gcd(A)=1\) and in such a case \(S\) is called a \textit{numerical semigroup}. The finite value \(\operatorname{g}(S)=|\mathbb{N}_0 \setminus S|\) is called the \textit{genus} of \(S\). One of the main research topics in this context is to find expressions of the invariants of a numerical semigroup \(S\), like the genus \(\operatorname{g}(S)\), in terms of the generators of \(S\). Another studied notion in this context is that of \textit{quotient} of a numerical semigroup \(S\) by a positive integer \(e\). In particular, it is defined as the numerical semigroup \(S/e=\{x/e\mid x\in S, x/e \in \mathbb{N}_0\}\). In the paper under review, the authors show expressions for the genus of the quotient of a numerical semigroup \(S\), in the cases where the set of generators of \(S\) is an arithmetic progression, a geometric progression or a Pythagorean triple. In particular, in order to obtain their results, the authors find the expression of the Hilbert series of the examined numerical semigroups. Taking advantage of this, the expression of \(\operatorname{g}(S/e)\) is shown for every positive integer \(e\), when \(S\) is generated by an arithmetic progression. The same goal is obtained when \(S\) is generated by a geometric progression or a Pythagorean triple and if \(e\) divides or is relatively prime to a specific generator of \(S\). For this type of semigroups, the authors propose as an open question the study of the case \(e\) neither divides nor is relatively prime to the specific generator of \(S\).