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{{Short description|Additional mathematical object}}
In [[mathematics]], a '''structure''' on a [[set]], or more generally a [[intuitionistic type theory|type]], consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.
{{About||the notion of "structure" in [[mathematical logic]]|Structure (mathematical logic)}}
{{more footnotes|date=April 2016}}


In Mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an [[Operation (mathematics)|operation]], [[Relation (mathematics)|relation]], [[Metric (mathematics)|metric]], or [[topological space|topology]]). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) 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]], [[equivalence relation]]s, and [[differential structure]]s.


⚫
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]], [[graph theory|graphs]], [[Event structure|events]], [[equivalence relation]]s, [[differential structure]]s, and [[Category (mathematics)|categories]].
Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an order induces a topology. As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a [[topological group]].


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 [[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}}</ref>
⚫
[[Map (mathematics)|Mappings]] between sets which preserve structures (so that structures in the domain are mapped to equivalent structures in the 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)|Map]] between two sets with the same type of structure, which preserve this structure [<nowiki/>[[morphism]]: structure in the [[Domain of a function|domain]] is mapped properly to the (same type) structure in the [[codomain]]] is of special interest in many fields of mathematics. Examples are [[homomorphisms]], which preserve algebraic structures; [[continuous functions]], which preserve topological structures; and [[differentiable functions]], 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.<ref name=Corry>{{cite journal|last1=Corry|first1=Leo|title=Nicolas Bourbaki and the concept of mathematical structure|journal=Synthese|date=September 1992|volume=92|issue=3|pages=315&ndash;348|jstor=20117057|doi=10.1007/bf00414286|s2cid=16981077}}</ref> They identified '''three ''mother structures'': algebraic, topological, and order'''.<ref name=Corry/><ref name=Wells>{{cite book|last1=Wells|first1=Richard B.|title=Biological signal processing and computational neuroscience|date=2010|pages=296&ndash;335|url=http://www.mrc.uidaho.edu/~rwells/techdocs/Biological%20Signal%20Processing/Chapter%2010%20Mathematical%20Structures.pdf|access-date=7 April 2016}}</ref>


==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 every other number.
*An order: each number is either less than or greater 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 addition and multiplication, the first of which makes it into a [[group theory |group]] and the pair of which together make it into a [[Field (mathematics)|field]].
*a measure: intervals along the real line have a certain [[length]].
*A measure: [[interval (mathematics)|intervals]] of the real line have a specific [[length]], which can be extended to the [[Lebesgue measure]] on many of its [[subset]]s.
*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|flat]].
*A geometry: it is equipped with a [[Metric (mathematics)|metric]] and is [[Flat space|flat]].
*a topology: there is a notion of open sets. (this is implied by the metric)
*A topology: there is a notion of [[open set]]s.
There are interfaces among these:
There are interfaces among these:
*Its order and, independently, its metrics structure induce its topology.
*Its order and, independently, its metric structure induce its topology.
*Its order and algebraic structure make it into an [[ordered field]].
*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]].
*Its algebraic structure and topology make it into a [[Lie group]], a type of [[topological group]].


== See also ==
== See also ==
⚫
*[[Abstract structure]]
*[[Isomorphism]]
*[[Equivalent definitions of mathematical structures]]
*[[Intuitionistic type theory]]
*[[Mathematical object]]
*[[Algebraic structure]]
*[[Space (mathematics)]]
*[[Category (mathematics)]]


== References ==
⚫
* [[Abstract structure]]
{{Reflist|}}


==References==
==Further reading==
*Bourbaki, Nikolas (1968). “Elements of Mathematics: Theory of Sets”. Hermann, Addison-Wesley. pp. 259-346, 383-385.
⚫
* {{planetmath reference|id=3017|title=Structure}} ''(provides a categorical definition.)''
*{{cite book|last1=Foldes|first1=Stephan|title=Fundamental Structures of Algebra and Discrete Mathematics|url=https://archive.org/details/fundamentalstruc0000fold|url-access=registration|date=1994|publisher=John Wiley & Sons|location=Hoboken|isbn=9781118031438}}
*{{cite journal|last1=Hegedus|first1=Stephen John|last2=Moreno-Armella|first2=Luis|title=The emergence of mathematical structures|journal=Educational Studies in Mathematics|date=2011|volume=77|issue=2|pages=369&ndash;388|doi=10.1007/s10649-010-9297-7|s2cid=119981368}}
*{{cite book|last1=Kolman|first1=Bernard|last2=Busby|first2=Robert C.|last3=Ross|first3=Sharon Cutler|title=Discrete mathematical structures|date=2000|publisher=Prentice Hall|location=Upper Saddle River, NJ|isbn=978-0-13-083143-9|edition=4th}}
*{{cite book|last1=Malik|first1=D.S.|last2=Sen|first2=M.K.|title=Discrete mathematical structures : theory and applications|date=2004|publisher=Thomson/Course Technology|location=Australia|isbn=978-0-619-21558-3}}
*{{cite book|last1=Pudlák|first1=Pavel|chapter=Mathematical structures|title=Logical foundations of mathematics and computational complexity a gentle introduction|date=2013|publisher=Springer|location=Cham|isbn=9783319001197|pages=2&ndash;24}}
*{{cite journal|last1=Senechal|first1=M.|author-link=Marjorie Senechal|title=Mathematical Structures|journal=Science|date=21 May 1993|volume=260|issue=5111|pages=1170–1173|doi=10.1126/science.260.5111.1170|pmid=17806355}}


==External links==
⚫
* {{planetmath reference|urlname=Structure|title=Structure}} ''(provides a model theoretic definition.)''
* [http://journals.cambridge.org/action/displayJournal?jid=MSC Mathematical structures in computer science] (journal)

{{Mathematical logic}}
{{Authority control}}

[[Category:Mathematical structures| ]]
[[Category:Type theory]]
[[Category:Type theory]]
[[Category:Set theory]]
[[Category:Set theory]]

[[de:Hierarchie mathematischer Strukturen]]
[[ka:მათემატიკური სტრუქტურა]]
[[hu:Matematikai struktúra]]
[[ja:数学的構造]]
[[pl:Struktura matematyczna]]
[[sv:Struktur (matematik)]]
[[ta:கணித அமைப்பு]]
[[zh:数学结构]]

Revision as of 08:24, 13 December 2024

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

In Mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.

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

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 topological group.[1]

Map between two sets with the same type of structure, which preserve this structure [morphism: structure in the domain is mapped properly to the (same type) structure in the codomain] is of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; continuous functions, which preserve topological structures; and differentiable functions, 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.[2] They identified three mother structures: algebraic, topological, and order.[2][3]

Example: the real numbers

The set of real numbers has several standard structures:

  • An order: each number is either less than or greater than any other number.
  • Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field.
  • A measure: intervals of 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. ^ Saunders, Mac Lane (1996). "Structure in Mathematics" (PDF). Philosoph1A Mathemat1Ca. 4 (3): 176.
  2. ^ 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. S2CID 16981077.
  3. ^ Wells, Richard B. (2010). Biological signal processing and computational neuroscience (PDF). pp. 296–335. Retrieved 7 April 2016.

Further reading

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