# Algorithms for Generalized Numerical Semigroups

## TL;DR

This paper introduces algorithms for analyzing generalized numerical semigroups in higher dimensions, enabling the computation of their structure, gaps, generators, and enumeration based on genus, advancing understanding of their combinatorial properties.

## Contribution

It presents new algorithms for identifying, analyzing, and enumerating generalized numerical semigroups in multiple dimensions, including a novel method to count all such semigroups with a given genus.

## Key findings

- Successfully computed the number of generalized numerical semigroups for various dimensions and genus.
- Developed algorithms to determine if a set generates a generalized numerical semigroup.
- Enabled enumeration of semigroups with prescribed genus, surpassing previous computational limits.

## Abstract

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in $\mathbb{N}^d$ we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in $\mathbb{N}^d$ we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). It was used to compute the number of such semigroups, and its implementation allowed us to compute (for various dimensions) the number of numerical semigroups for genus that had not been attained before.