# Irreducible Generalized Numerical Semigroups and uniqueness of the Frobenius element

## TL;DR

This paper introduces irreducible generalized numerical semigroups in higher dimensions, characterizes them via a special subset, and proves the Frobenius element's uniqueness regardless of the monomial order for these semigroups.

## Contribution

It defines irreducibility for generalized numerical semigroups, characterizes them, and establishes the order-independence of the Frobenius element in this context.

## Key findings

- Irreducible generalized numerical semigroups are characterized by a specific subset of their gaps.
- The Frobenius element is independent of the relaxed monomial order for irreducible semigroups.
- The paper introduces relaxed monomial orders and their relation to the Frobenius element.

## Abstract

Let $\mathbb{N}^{d}$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $ S\subseteq \mathbb{N}^d$ such that $H(S)=\mathbb{N}^d \setminus S$ is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of $H(S)$. In particular, we describe relaxed monomial orders on $\mathbb N^d$, define the Frobenius element of $S$ with respect to a given relaxed monomial order, and show that the Frobenius element of $S$ is independent of the order if the generalized numerical semigroup is irreducible.