In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain.
Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure.
In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are at most those induced from the bigger structure.
In the language (×, 1) of monoids, however, the substructures of a group are its submonoids.
In model theory, given a structure M which is a model of a theory T, a submodel of M in a narrower sense is a substructure of M which is also a model of T. For example, if T is the theory of abelian groups in the signature (+, 0), then the submodels of the group of integers (Z, +, 0) are the substructures which are also abelian groups.