Difference between revisions of "Mathematical structure"
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As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group. | As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group. | ||
| − | == | + | == Mapping and preservation of structure == |
Mappings between sets which preserve structures (so that 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. | Mappings between sets which preserve structures (so that 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. | ||
Revision as of 19:06, 4 May 2016
In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance. (See also type.)
Contents
Description
A partial list of structures includes:
- Measures
- Algebraic structures (groups, fields, etc.)
- Topologies
- Metric structures (geometries)
- Orders
- Events
- Equivalence relations
- Differential structures
- 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 rigid form, shape, or topology on the set.
As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group.
Mapping and preservation of structure
Mappings between sets which preserve structures (so that 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; and diffeomorphisms, which preserve differential structures.
Bourbaki
N. Bourbaki suggested an explicit concept of "mathematical structure" in their book "Theory of Sets" (Chapter 4. Structures) and then defined, on that basis, a very general concept of isomorphism.
See also
- Abstract structure
- Algebraic structure
- Equivalent definitions of mathematical structures
- Intuitionistic type theory
- Mathematics
- Space (mathematics)
External links
- Mathematical structure @ Wikipedia
