On the number of generalized numerical semigroups

TL;DR

This paper establishes new bounds on the number of d-dimensional generalized numerical semigroups, extending classical concepts and introducing partition labelings to analyze their structure.

Contribution

It provides the best known bounds on the count of generalized numerical semigroups and extends key notions like multiplicity and depth to this setting.

Findings

Derived explicit bounds involving roots of specific polynomials

Extended multiplicity and depth concepts to generalized semigroups

Introduced partition labelings generalizing Kunz words

Abstract

Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}} \mathsf{r}_{2^d}^g\] for some constant $C_d > 0$, which can be made explicit. To do this, we extend the notion of multiplicity and depth to generalized numerical semigroups and show our lower bound is sharp for semigroups of depth 2. We also show other bounds on special classes of semigroups by introducing partition labelings, which extend the notion of Kunz words to the general setting.