> 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/900-index/mathematical/mathematical-structure.md). # 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 )