Substructure - Submodel

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 groups. Thus the natural numbers (N, +, 0) form a substructure of (Z, +, 0) which is not a submodel, while the even numbers (2Z, +, 0) form a submodel which is (a group but) not a subgroup.

Other examples:

1. The algebraic numbers form a submodel of the complex numbers in the theory of algebraically closed fields.
2. The rational numbers form a submodel of the real numbers in the theory of fields.
3. Every elementary substructure of a model of a theory T also satisfies T; hence it is a submodel.

In the category of models of a theory and embeddings between them, the submodels of a model are its subobjects.