Mathematical structure


Mathematical structure

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.

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

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 homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.

N. Bourbaki suggested an explication of the concept "mathematical structure" in their book "Theory of Sets" (Chapter 4. Structures) and then defined on that base, in particular, a very general concept of isomorphism.

Example: the real numbers

The set of real numbers has several standard structures:

  • an order: each number is either less or more than every 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 certain 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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • mathematical structure — noun A set together with a collection of mathematical objects such as other sets, relations or operations which endow it with additional stucture …   Wiktionary

  • Structure — is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities. From a child s verbal description of a snowflake, to the detailed scientific analysis of the… …   Wikipedia

  • Mathematical universe hypothesis — In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the Ultimate Ensemble, is a speculative theory of everything (TOE) proposed by the theoretical physicist, Max Tegmark.[1] Contents 1 Description 2 Criticisms and… …   Wikipedia

  • Mathematical formulation of quantum mechanics — Quantum mechanics Uncertainty principle …   Wikipedia

  • Structure (disambiguation) — The structure of a thing is how the parts of it relate to each other, how it is put together .Structure may also refer to:* architectural structure, a man made structure used or intended for supporting or sheltering any use or continuous… …   Wikipedia

  • Mathematical economics — Economics …   Wikipedia

  • Structure (mathematics) — In mathematics, structure can refer to: * Mathematical structure ndash; extra objects attached to a set that endow it with extra meaning or significance. * A structure in mathematical logic, sometimes called a model, gives a concrete… …   Wikipedia

  • Mathematical logic — (also known as symbolic logic) is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic.[1] The field includes both the mathematical study of logic and the… …   Wikipedia

  • Mathematical Association — Abbreviation MA Formation 1871 Legal status Non profit organisation and registered charity Purpose/focus Professional organisation for mathematics educators Location …   Wikipedia

  • Mathematical and theoretical biology — is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology.[1] The field may be referred to as mathematical biology or biomathematics to stress the mathematical side, or as theoretical… …   Wikipedia