> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/400-abstract-algebra/terms/mathematical-structure.md).

# Mathematical structure

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

In abstract algebra, a **mathematical structure** is a set, referred to as a **carrier set**, endowed with one or more additional features, such as operations, relations, orders, metrics, topologies. Almost always, these extra features, whether attached or just related to the carrier set, endow it with some additional meaning.

## List of mathematical structures

Mathematical structures include

* measures
* algebraic structures (groups, fields...)
* topologies
* metric structures (geometries)
* orders
* events
* equivalence relations
* differential structures
* categories

Sometimes, a set is endowed with more than one structure 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 structure and a group structure, such that these two structures are related in a certain way, then the set becomes a *topological group*.

Mappings between sets which preserve structures (i.e. 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:

* *homomorphisms* preserve algebraic structures
* *homeomorphisms* preserve topological structures
* *diffeomorphisms* preserve differential structures