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

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.