# Herbrand structure

In mathematics, for a language $\mathcal{L}$, define the Herbrand universe to be the set of ground terms of $\mathcal{L}$.
A structure $\mathfrak{M}$ for $\mathcal{L}$ is a Herbrand structure if the domain of $\mathfrak{M}$ is the Herbrand universe of $\mathcal{L}$ and the interpretation of $\mathfrak{M}$ is a Herbrand interpretation. This fixes the domain of $\mathfrak{M}$, and so each Herbrand structure can be identified with its interpretation.
A Herbrand model of a theory $T$ is a Herbrand structure which is a model of $T.$