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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a structure on a set, or more generally a type
Intuitionistic type theory

Intuitionistic type theory, or constructive type theory, or Martin-L?f type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism....
, consists of additional mathematical object
Mathematical object

In mathematics and its philosophy of mathematics, a mathematical object is an abstract object arising in mathematics. Commonly encountered mathematical objects include numbers, permutations, Partition of a set, matrix , set , function , and relation ....
s that in some manner attach (or are related) 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 structure
Algebraic structure

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more Set Closure under one or more Operation , satisfying some axiom....
s (group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
s, field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
s, etc.), topologies
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, metric structures
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
 (geometries
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
), orders
Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
, equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
s, and differential structure
Differential structure

In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifold, which is a Topological manifold with some additional structure that allows us to do differential calculus on the manifold....
s.

Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a structure on a set, or more generally a type
Intuitionistic type theory

Intuitionistic type theory, or constructive type theory, or Martin-L?f type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism....
, consists of additional mathematical object
Mathematical object

In mathematics and its philosophy of mathematics, a mathematical object is an abstract object arising in mathematics. Commonly encountered mathematical objects include numbers, permutations, Partition of a set, matrix , set , function , and relation ....
s that in some manner attach (or are related) 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 structure
Algebraic structure

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more Set Closure under one or more Operation , satisfying some axiom....
s (group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
s, field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
s, etc.), topologies
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, metric structures
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
 (geometries
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
), orders
Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
, equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
s, and differential structure
Differential structure

In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifold, which is a Topological manifold with some additional structure that allows us to do differential calculus on the manifold....
s.

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
Topological group

In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
.

Mappings
Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for Function . Thus, for example, a partial map is a partial function, and a total map is a total function....
 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 homomorphism
Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
s, which preserve algebraic structures; homeomorphism
Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
s, which preserve topological structures; and diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
s, which preserve differential structures.

Example: the real numbers

The set of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s 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
    Field (mathematics)

    In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
    .
  • a measure: intervals along the real line have a certain length
    Length

    Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end....
    , which can be extended to the Lebesgue measure
    Lebesgue measure

    In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
     on many of its subsets.
  • a metric: there is a notion of distance
    Metric (mathematics)

    In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
     between points.
  • a geometry: it is equipped with a metric
    Metric (mathematics)

    In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
     and is flat
    Flatness

    The intuitive idea of flatness is important in several fields....
    .
  • 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
    Ordered field

    In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations....
    .
  • Its algebraic structure and topology make it into a Lie group
    Lie group

    In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
    , a type of topological group
    Topological group

    In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
    .


See also

  • Structure (mathematical logic)
    Structure (mathematical logic)

    In universal algebra and in model theory, a structure consists of an underlying Set along with a collection of finitary functions and relations which are defined on it....
  • Abstract algebra
    Abstract algebra

    Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
  • Abstract structure
    Abstract structure

    An abstract structure is a formal object that is defined by a set of laws, properties, and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects....