Algorithms and basic asymptotics for generalized numerical semigroups in ${\mathbb {N}}^d$

Abstract

Let ${\mathbb {N}}$ denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid $S\subseteq {\mathbb {N}}$. For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid $S\subseteq {\mathbb {N}}^d$. The cardinality of ${\mathbb {N}}^d \setminus S$ is called the genus. We describe a family of algorithms, parameterized by (relaxed) monomial orders, that can be used to generate trees of semigroups with each GNS appearing exactly once. Let $N_{g,d}$ denote the number of generalized numerical semigroups $S\subseteq {\mathbb {N}}^d$ of genus $g$. We compute $N_{g,d}$ for small values of $g,d$ and provide coarse asymptotic bounds on $N_{g,d}$ for large values of $g,d$. For a fixed $g$, we show that $F_g(d)=N_{g,d}$ is a polynomial function of degree $g$. We close with several open problems/conjectures related to the asymptotic growth of $N_{g,d}$ and with suggestions for further avenues of research.