Frobenius allowable gaps of Generalized Numerical Semigroups

TL;DR

This paper characterizes Frobenius allowable gaps in generalized numerical semigroups, introduces the concept of Frobenius GNS, and estimates their abundance, revealing new structural insights in higher dimensions.

Contribution

It provides a characterization of Frobenius allowable gaps, introduces Frobenius GNS, and estimates their count, advancing understanding of the structure of generalized numerical semigroups.

Findings

Characterization of Frobenius allowable gaps in GNS.

Introduction of Frobenius GNS with a single Frobenius gap.

Asymptotic estimate of the number of Frobenius GNS for large dimensions.

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.