by restricting all functions and relations in ? to the subset.This generalises the analogous concepts from algebra; for instance, a subgroup is a substructure in the signature with multiplication and inverse.

A substructure is said to be elementary if for any first-order formula and any elements a₁, ..., a_n of ,
: if and only if .In particular, if is a sentence and an elementary substructure of , then if and only if . Thus, an elementary substructure is a model