Symmetric generalized numerical semigroups in N^d with embedding dimension 2d+1

In this article, we classify all symmetric generalized numerical semigroups in Nd of embedding dimension 2d + 1. Consequently, we show that in the case d > 1, the property of being symmetric is equivalent to have a unique maximal gap with respect to natural partial order in Nd . 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.