Mathematical structure: Difference between revisions

Content deleted Content added

Inline

Revision as of 11:46, 9 March 2016

In mathematics, a structure on a set is additional mathematical objects that, in some manner, attach (or relate) to that set to endow it with meaning or significance. (See also type)

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; this enables mathematicians to study it more richly. For example, an ordering imposes a ridged form, shape, or topology on the set. As another example, if a set both has a topology and is a group, and these 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 source or domain are mapped to equivalent structures in the destination or codomain, for example ordering) 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 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.

References

"Structure". PlanetMath. (provides a model theoretic definition.)
D.S. Malik and M. K. Sen (2004) Discrete mathematical structures: theory and applications, ISBN 978-0-619-21558-3 .
M. Senechal (1993) "Mathematical Structures", Science 260:1170–3.
Bernard Kolman, Robert C. Ross, and Sharon Cutler (2004) Discrete mathematical Structures, ISBN 978-0-13-083143-9 .
Stephen John Hegedes and Luis Moreno-Armella (2011)"The emergence of mathematical structures", Educational Studies in Mathematics 77(2):369–88.
Journal: Mathematical structures in computer science, Cambridge University Press ISSN 0960-1295.

@@ Line 5: / Line 5: @@
 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]], [[Event structure|events]], [[equivalence relation]]s, [[differential structure]]s, and [[Category (category theory)|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 structure simultaneously; this enables mathematicians to study it more richly. For example, an ordering imposes a ridged form, shape, or topology on the set. As another example, if a set both has a topology and is a group, and these two structures are related in a certain way, the set becomes a [[topological group]].
-[[Map (mathematics)|Mappings]] between sets which preserve structures (so that structures in the [[Domain of a function|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)|Mappings]] between sets which preserve structures (so that structures in the source or [[Domain of a function|domain]] are mapped to equivalent structures in the destination or [[codomain]], for example ordering) 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.
 [[Nicolas Bourbaki|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.