Mathematical structure

In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.

A partial list of possible structures is measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, graphs, events, differential structures, categories, setoids, and equivalence relations.

Sometimes, a set is endowed with more than one feature simultaneously, which allows to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on a set, and if a set has both a topology feature and a group feature, such that these two features are compatible in a certain way, then the structure becomes a topological group.

A map between two similarly-structured sets that preserves their structure is known as a morphism, and such maps are of special interest in many fields of mathematics. Examples include homomorphisms, which preserve algebraic structures; continuous functions, which preserve topological structures; and differentiable functions, which preserve differential structures. Morphisms that can be inverted, known as isomorphisms, allow to describe when sets endowed with the same kind of structure have the same properties from the point of view of that structure, in which case they are called isomorphic.

History

In 1939, the French group with the pseudonym "Nicolas Bourbaki" saw structures as the root of mathematics.