mathematics, a structure is a
set endowed with some additional features on the set (e.g. an
topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.
Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a
Mappings between sets which preserve structures (i.e., structures in the
domain are mapped to equivalent structures in the
codomain) are of special interest in many fields of mathematics. Examples are
homomorphisms, which preserve algebraic structures;
homeomorphisms, which preserve topological structures; and
diffeomorphisms, which preserve differential structures.
In 1939, the French group with the pseudonym
Nicolas Bourbaki saw structures as the root of mathematics. They first mentioned them in their "Fascicule" of Theory of Sets and expanded it into Chapter IV of the 1957 edition. They identified three mother structures: algebraic, topological, and order.