A semigroup defining the Gr\"obner degeneration of a toric ideal

We give an explicit set of generators for the semigroup of the Gr"obner degeneration of a toric ideal. This set of generators is used to study algebraic properties of the semigroup it generates: approximation of semigroups, non-preservation of saturation, Betti elements, uniqueness of presentations, and M"obius functions.