Substructure
In mathematical logic, an
substructure or
subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an
extension or a
superstructure of its substructure. In model theory, the term
"submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models. In the presence of relations it may make sense to relax the conditions on a subalgebra so that the relations on a
weak substructure are
at most those induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures. Ordered groups, on the other hand, have the special property that every substructure of an ordered group which is itself an ordered group, is an induced substructure.