Frobenius allowable gaps of Generalized Numerical Semigroups

Abstract

A generalised numerical semigroup (GNS) is a submonoid $S$ of ${\mathbb{N}}^d$ for which the complement ${\mathbb{N}}^d\setminus S$ is finite. The points in the complement ${\mathbb{N}}^d\setminus S$ are called gaps. A gap $F$ is considered Frobenius allowable if there is some relaxed monomial ordering on ${\mathbb{N}}^d$ with respect to which $F$ is the largest gap. We characterise the Frobenius allowable gaps of a GNS. A GNS that has only one Frobenius allowable gap is called a Frobenius GNS. We estimate the number of Frobenius GNS with a given Frobenius gap $F=(F^{(1)},\dots ,F^{(d)})\in {\mathbb{N}}^d$ and show that it is close to ${\sqrt{3}}^{(F^{(1)}+1)\cdots (F^{(d)}+1)}$ for large $d$. We define notions of quasi-irreducibility and quasi-symmetry for GNS. While in the case of $d=1$ these notions coincide with irreducibility and symmetry of GNS, they are distinct in higher dimensions.

20 pages