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Concept
Substructure (mathematics)
Summary
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. 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 (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. 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.
Given two structures A and B of the same signature σ, A is said to be a weak substructure of B, or a weak subalgebra of B, if
the domain of A is a subset of the domain of B,
f A = f B|An for every n-ary function symbol f in σ, and
R A R B An for every n-ary relation symbol R in σ.
A is said to be a substructure of B, or a subalgebra of B, if A is a weak subalgebra of B and, moreover,
R A = R B An for every n-ary relation symbol R in σ.
If A is a substructure of B, then B is called a superstructure of A or, especially if A is an induced substructure, an extension of A.
In the language consisting of the binary functions + and ×, binary relation
Official source
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