Mathematical structure

A set or collection equipped with specified relations, operations, or other features that determine its mathematical organization. The same underlying set can support different structures.

Set: A collection of distinct objects treated as a single mathematical object. A structure begins with an underlying set of objects.
Algebraic structure: A set equipped with operations satisfying specified identities. Groups, rings, and similar objects are structures defined chiefly by operations.
Unstructured set: A set considered without any designated relations, operations, or additional features. It isolates the bare collection from the organization added by a structure.
Cayley's theorem: The theorem that every group is isomorphic to a subgroup of a permutation group. It realizes an abstract group structure through permutations.
Relation: A specification of which tuples of elements satisfy a given condition. Relations can encode order, adjacency, or equivalence on the underlying set.
Order theory: The study of partially and totally ordered sets and their properties. Order relations are a central kind of mathematical structure.
Isomorphism: A structure-preserving map with an inverse that also preserves structure. Isomorphic structures differ in presentation but have the same organization.
Graph theory: The study of vertices and edges and the relationships they encode. Graphs model relational structure in networks, geometry, and computation.
Binary operation: A rule that assigns an element of a set to each ordered pair of its elements. Operations add algebraic behavior to a set.
Topology: The study of properties preserved under continuous deformation, formalized through open sets or equivalent data. Topological spaces carry structure through designated collections of open subsets.

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