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{{more footnotes|date=April 2016}}
{{more footnotes|date=April 2016}}


In [[mathematics]], a '''structure''' on a [[Set (mathematics)|set]] is an additional [[mathematical object]] that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.
In [[mathematics]], a '''structure''' is a set endowed with some additional features on the set (e.g., [[Operation (mathematics)|operation]], [[Relation (math)|relation]], [[Metric (mathematics)|metric]], [[Topology#Topologies on sets|topology]]).<ref>{{Cite web|url=https://mathvault.ca/math-glossary/#structure|title=The Definitive Glossary of Higher Mathematical Jargon — Mathemaical Structure|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-12-09}}</ref> Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.


A partial list of possible structures are [[Measure theory|measures]], [[algebraic structure]]s ([[group (mathematics)|group]]s, [[field (mathematics)|field]]s, etc.), [[Topology|topologies]], [[Metric space|metric structures]] ([[Geometry|geometries]]), [[Order theory|orders]], [[Event structure|events]], [[equivalence relation]]s, [[differential structure]]s, and [[Category (category theory)|categories]].
A partial list of possible structures are [[Measure theory|measures]], [[algebraic structure]]s ([[group (mathematics)|group]]s, [[field (mathematics)|field]]s, etc.), [[Topology|topologies]], [[Metric space|metric structures]] ([[Geometry|geometries]]), [[Order theory|orders]], [[Event structure|events]], [[equivalence relation]]s, [[differential structure]]s, and [[Category (category theory)|categories]].


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]].
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]].<ref>{{Cite journal|last=Saunders|first=Mac Lane|date=1996|title=Structure in Mathematics|url=http://www2.mat.ulaval.ca/fileadmin/Pages_personnelles_des_profs/hm/H14_Mac_Lane_Phil_Math_1996.pdf|journal=PHILOSOPH1A MATHEMAT1CA|volume=4|issue=3|pages=176|via=}}</ref>


[[Map (mathematics)|Mappings]] between sets which preserve structures (so that structures in the source or [[Domain of a function|domain]] are mapped to equivalent structures in the destination or [[codomain]]) are of special interest in many fields of mathematics. Examples are [[homomorphism]]s, which preserve algebraic structures; [[homeomorphism]]s, which preserve topological structures; and [[diffeomorphism]]s, which preserve differential structures.
[[Map (mathematics)|Mappings]] between sets which preserve structures (i.e., structures in the source or [[Domain of a function|domain]] are mapped to equivalent structures in the destination or [[codomain]]) are of special interest in many fields of mathematics. Examples are [[homomorphism]]s, which preserve algebraic structures; [[homeomorphism]]s, which preserve topological structures;<ref>{{Cite web|url=http://www.maths.lth.se/matematiklth/personal/stordal/kompendium.pdf|title=Mathematical structures|last=Christiansen|first=Jacob Stordal|date=2015|website=maths.lth.se|url-status=live|archive-url=|archive-date=|access-date=2019-12-09}}</ref> and diffeomorphisms, which preserve differential structures.


==History==
==History==
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==Example: the real numbers==
==Example: the real numbers==
The set of [[real number]]s has several standard structures:
The set of [[real number]]s has several standard structures:
*an order: each number is either less or more than any other number.
*An order: each number is either less or more than any other number.
*algebraic structure: there are operations of multiplication and addition that make it into a [[Field (mathematics)|field]].
*Algebraic structure: there are operations of multiplication and addition that make it into a [[Field (mathematics)|field]].
*a measure: intervals along the real line have a specific [[length]], which can be extended to the [[Lebesgue measure]] on many of its subsets.
*A measure: intervals along the real line have a specific [[length]], which can be extended to the [[Lebesgue measure]] on many of its subsets.
*a metric: there is a notion of [[Metric (mathematics)|distance]] between points.
*A metric: there is a notion of [[Metric (mathematics)|distance]] between points.
*a geometry: it is equipped with a [[Metric (mathematics)|metric]] and is [[Flatness (mathematics)|flat]].
*A geometry: it is equipped with a [[Metric (mathematics)|metric]] and is [[Flatness (mathematics)|flat]].
*a topology: there is a notion of open sets.
*A topology: there is a notion of open sets.
There are interfaces among these:
There are interfaces among these:
*Its order and, independently, its metric structure induce its topology.
*Its order and, independently, its metric structure induce its topology.

Revision as of 22:24, 9 December 2019

For the notion of "structure" in mathematical logic, see Structure (mathematical logic). For structures in category theory, see Structure (category theory).

In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology).[1] Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and 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.[2]

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. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures;[3] and diffeomorphisms, which preserve differential structures.

History

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.[4] They identified three mother structures: algebraic, topological, and order.[4][5]

Example: the real numbers

The set of real numbers has several standard structures:

  • An order: each number is either less or more than any other number.
  • Algebraic structure: there are operations of multiplication and addition that make it into a field.
  • A measure: intervals along the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.
  • A metric: there is a notion of distance between points.
  • A geometry: it is equipped with a metric and is flat.
  • A topology: there is a notion of open sets.

There are interfaces among these:

  • Its order and, independently, its metric structure induce its topology.
  • Its order and algebraic structure make it into an ordered field.
  • Its algebraic structure and topology make it into a Lie group, a type of topological group.

See also

References

  1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Mathemaical Structure". Math Vault. 2019-08-01. Retrieved 2019-12-09.{{cite web}}: CS1 maint: url-status (link)
  2. ^ Saunders, Mac Lane (1996). "Structure in Mathematics" (PDF). PHILOSOPH1A MATHEMAT1CA. 4 (3): 176.
  3. ^ Christiansen, Jacob Stordal (2015). "Mathematical structures" (PDF). maths.lth.se. Retrieved 2019-12-09.{{cite web}}: CS1 maint: url-status (link)
  4. ^ a b Corry, Leo (September 1992). "Nicolas Bourbaki and the concept of mathematical structure". Synthese. 92 (3): 315–348. doi:10.1007/bf00414286. JSTOR 20117057.
  5. ^ Wells, Richard B. (2010). Biological signal processing and computational neuroscience (PDF). pp. 296–335. Retrieved 7 April 2016.

Further reading

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