Mathematical structure

https://en.wikipedia.org/wiki/Mathematical_structure

In abstract algebra, a mathematical structure is a set, referred to as a carrier or underlying set, endowed with one or more additional features (often also called structures), such as operations, relations, orders. Almost always, these extra features, whether attached or related to the carrier set, endow it with some additional meaning.

A structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional significance.

Mathematical structures include

• measures

• algebraic structures: groups, rings, fields, modules, vector spaces, algebras

• topological structures, topologies

• differential structures

• metric structures, geometries

• orders

• events

• equivalence relations

• categories

• Euclidean space

Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an ordering imposes a rigid form, shape or topology on the set. As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group.

Mappings between sets which preserve structures (so that structures in the source or domain are mapped to equivalent structures in the destination or codomain) are of special interest in many fields of mathematics. Examples are:

• homomorphisms preserve algebraic structures

• homeomorphisms preserve topological structures

• diffeomorphisms preserve differential structures

References

https://en.wikipedia.org/wiki/Mathematical_structure https://en.wikipedia.org/wiki/Structure_(mathematical_logic) https://en.wikipedia.org/wiki/Algebraic_structure https://en.wikipedia.org/wiki/Abstract_structure

https://en.wikipedia.org/wiki/Category:Mathematical_structures https://en.wikipedia.org/wiki/Equivalent_definitions_of_mathematical_structures