mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a structure on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

(or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation,

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

, or

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance. A partial list of possible structures are measures,

algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...

s ( groups, fields, etc.), topologies, metric structures (

geometries Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Types, methodologies, and terminologies of geometry. ...

), orders, graphs, events,

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

differential structure In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for ...

s, and categories. Sometimes, a set is endowed with more than one feature 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 feature and a group feature, such that these two features are related in a certain way, then the structure becomes a

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

Map A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...

between two sets with the same type of structure, which preserve this structure nowiki/>morphism: structure in the domain is mapped properly to the (same type) structure in the Domain of a function">domain is mapped properly to the (same type) structure in the codomain">codomain">Domain of a function">domain is mapped properly to the (same type) structure in the codomainis of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; continuous functions, which preserve topological structures; and differentiable functions, which preserve differential structures.

History

In 1939, the French group with the pseudonym "

Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intende ...

" 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. They identified three ''mother structures'': algebraic, topological, and order.

Example: the real numbers

The set of

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s has several standard structures: *An order: each number is either less than or greater than any other number. *Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field. *A measure: intervals of the real line have a specific

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

, which can be extended to the

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

on many of its

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

s. *A metric: there is a notion of

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

between points. *A geometry: it is equipped with a

and is flat. *A topology: there is a notion of

open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

s. 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 In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...

. *Its algebraic structure and topology make it into a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

, a type of

References

External links

* ''(provides a model theoretic definition.)''
Mathematical structures in computer science
(journal) {{Authority control Type theory Set theory

History

Example: the real numbers

See also

References

Further reading

External links