Mathematical structure

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.

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

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.
  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.
  4. 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
  • 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.
  • 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.
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