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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.
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. They identified three mother structures: algebraic, topological, and order.
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
- Abstract structure
- Isomorphism
- Equivalent definitions of mathematical structures
- Intuitionistic type theory
- Mathematical object
- Algebraic structure
- Space (mathematics)
- Category (mathematics)
References
- Saunders, Mac Lane (1996). "Structure in Mathematics" (PDF). Philosoph1A Mathemat1Ca. 4 (3): 176.
- 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.
Further reading
- Bourbaki, Nikolas (1968). "Elements of Mathematics: Theory of Sets". Hermann, Addison-Wesley. pp. 259–346, 383–385.
- Foldes, Stephan (1994). Fundamental Structures of Algebra and Discrete Mathematics. Hoboken: John Wiley & Sons. ISBN 9781118031438.
- Hegedus, Stephen John; Moreno-Armella, Luis (2011). "The emergence of mathematical structures". Educational Studies in Mathematics. 77 (2): 369–388. doi:10.1007/s10649-010-9297-7. S2CID 119981368.
- Kolman, Bernard; Busby, Robert C.; Ross, Sharon Cutler (2000). Discrete mathematical structures (4th ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-083143-9.
- Malik, D.S.; Sen, M.K. (2004). Discrete mathematical structures : theory and applications. Australia: Thomson/Course Technology. ISBN 978-0-619-21558-3.
- Pudlák, Pavel (2013). "Mathematical structures". Logical foundations of mathematics and computational complexity a gentle introduction. Cham: Springer. pp. 2–24. ISBN 9783319001197.
- Senechal, M. (21 May 1993). "Mathematical Structures". Science. 260 (5111): 1170–1173. doi:10.1126/science.260.5111.1170. PMID 17806355.
External links
- "Structure". PlanetMath. (provides a model theoretic definition.)
- Mathematical structures in computer science (journal)
This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations April 2016 Learn how and when to remove this message 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 The 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 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 HistoryIn 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 They identified three mother structures algebraic topological and order Example the real numbersThe 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 alsoAbstract structure Isomorphism Equivalent definitions of mathematical structures Intuitionistic type theory Mathematical object Algebraic structure Space mathematics Category mathematics ReferencesSaunders Mac Lane 1996 Structure in Mathematics PDF Philosoph1A Mathemat1Ca 4 3 176 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 Further readingBourbaki Nikolas 1968 Elements of Mathematics Theory of Sets Hermann Addison Wesley pp 259 346 383 385 Foldes Stephan 1994 Fundamental Structures of Algebra and Discrete Mathematics Hoboken John Wiley amp Sons ISBN 9781118031438 Hegedus Stephen John Moreno Armella Luis 2011 The emergence of mathematical structures Educational Studies in Mathematics 77 2 369 388 doi 10 1007 s10649 010 9297 7 S2CID 119981368 Kolman Bernard Busby Robert C Ross Sharon Cutler 2000 Discrete mathematical structures 4th ed Upper Saddle River NJ Prentice Hall ISBN 978 0 13 083143 9 Malik D S Sen M K 2004 Discrete mathematical structures theory and applications Australia Thomson Course Technology ISBN 978 0 619 21558 3 Pudlak Pavel 2013 Mathematical structures Logical foundations of mathematics and computational complexity a gentle introduction Cham Springer pp 2 24 ISBN 9783319001197 Senechal M 21 May 1993 Mathematical Structures Science 260 5111 1170 1173 doi 10 1126 science 260 5111 1170 PMID 17806355 External links Structure PlanetMath provides a model theoretic definition Mathematical structures in computer science journal
