# On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$

## TL;DR

This paper explores submonoids of  with pseudo-Frobenius sets, generalizing numerical semigroups, and introduces a new family of submonoids related to affine semigroups and their invariants.

## Contribution

It characterizes submonoids of  with pseudo-Frobenius sets, generalizes invariants of numerical semigroups, and introduces a new family of submonoids as direct limits of affine semigroups.

## Key findings

- Submonoids with pseudo-Frobenius sets are affine semigroups with maximal projective dimension.
- Most invariants of numerical semigroups extend to these submonoids.
- A new family of submonoids is shown to be a direct limit of affine semigroups.

## Abstract

In this paper we study those submonoids of $\mathbb{N}^d$ which a non-trivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of $\mathbb{N}^d$ and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.