In first-order logic, a Herbrand structure S is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ.
Herbrand structures play an important role in the foundations of logic programming.
The relation symbols are not relevant for a Herbrand universe.
we have the following interpretation: In addition to the universe, defined in § Herbrand universe, and the term denotations, defined in § Herbrand structure, the Herbrand base completes the interpretation by denoting the relation symbols.
A Herbrand base is the set of all ground atoms whose argument terms are elements of the Herbrand universe.