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.
- The algebraic numbers form a submodel of the complex numbers in the theory of algebraically closed fields.
- The rational numbers form a submodel of the real numbers in the theory of fields.
- 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.
Read more about this topic: Substructure
Other articles related to "submodels, submodel":
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