Substructure

Substructure

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 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 (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.

Read more about Substructure:  Definition, Example, Substructures As Subobjects, Submodel, See Also

Other articles related to "substructure":

Structure (mathematical Logic) - Induced Substructures and Closed Subsets - Examples
... When regarded as σ-structures in the natural way, the rational numbers form a substructure of the real numbers, and the real numbers form a substructure of the ... The rational numbers are the smallest substructure of the real (or complex) numbers that also satisfies the field axioms ... The set of integers gives an even smaller substructure of the real numbers which is not a field ...
Substructure - See Also
... Elementary substructure End extension Löwenheim-Skolem theorem Prime model. ...
Entity–attribute–value Model - Representing Substructure: EAV With Classes and Relationships (EAV/CR)
... representation of highly diverse data, it is possible that a given object (class instance) may have substructure that is, some of its attributes may represent ... (The permissible substructure for a given class is defined within the system's attribute metadata, as discussed later ... apply to the class "computer" but not to the class "engine".) To represent substructure, we use a special kind of EAV table where the value column ...
Elementary Equivalence - Elementary Substructures and Elementary Extensions
... N is an elementary substructure of M if N and M are structures of the same signature σ such that for all first-order σ-formulas φ(x1, …, xn) with free variables x1, …, xn, and all elements a1 ... It follows that N is a substructure of M ... If N is a substructure of M, then both N and M can be interpreted as structures in the signature σN consisting of σ together with a new constant symbol for every element of N ...
Pyramid Of Neferefre - Pyramid - Substructure
... the devastation the Czech group found the funerary goods and even the king's mummy, all in the substructure ...