Mathematical structure: Difference between revisions

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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.

References

^ Saunders, Mac Lane (1996). "Structure in Mathematics" (PDF). Philosoph1A Mathemat1Ca. 4 (3): 176.
^ ^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.
^ Wells, Richard B. (2010). Biological signal processing and computational neuroscience (PDF). pp. 296–335. Retrieved 7 April 2016.

External links

"Structure". PlanetMath. (provides a model theoretic definition.)
Mathematical structures in computer science (journal)

[1] Saunders, Mac Lane (1996). "Structure in Mathematics" (PDF). Philosoph1A Mathemat1Ca. 4 (3): 176.

[Corry-2] 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.

[Wells-3] Wells, Richard B. (2010). Biological signal processing and computational neuroscience (PDF). pp. 296–335. Retrieved 7 April 2016.

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 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]] between two sets with the same type of structure, which preserve this structure (<nowiki>[[morphism]]</nowiki>: 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.
+[[Map (mathematics)|Map]] between two sets with the same type of structure, which preserve this structure (<nowiki>[[morphism]]</nowiki>: 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==