Symmetric generalized numerical semigroups in $\mathbb{N}^d$ with embedding dimension $2d+1$

TL;DR

This paper classifies symmetric generalized numerical semigroups in multi-dimensional space with a specific embedding dimension, revealing their unique properties and the non-existence of almost symmetric cases for higher dimensions.

Contribution

It provides a complete classification of symmetric generalized numerical semigroups with embedding dimension 2d+1 in any dimension d, establishing their equivalence to having a unique maximal gap.

Findings

Symmetric generalized numerical semigroups are classified for embedding dimension 2d+1.

Symmetry is equivalent to having a unique maximal gap in these semigroups.

No almost symmetric but not symmetric semigroups exist when d>1.

Abstract

In this article, we classify all symmetric generalized numerical semigroups in $\mathbb{N}^d$ of embedding dimension $2d+1$. Consequently, we show that in this case the property of being symmetric is equivalent to have a unique maximal gap with respect to natural partial order in $\mathbb{N}^d$. Moreover, we deduce that when $d>1$, there does not exist any generalized numerical semigroup of embedding dimension $2d+1$, which is almost symmetric but not symmetric.