In

Mathematical structures in computer science

(journal) {{Authority control Type theory Set theory

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

, or topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.
A partial list of possible structures are measures, algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s (group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

s, field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

s, etc.), topologies
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...

, metric structures (), orders
Orders is a surname
In some cultures, a surname, family name, or last name is the portion of one's personal name
300px, First/given, middle and last/family/surname with John Fitzgerald Kennedy as example. This shows a structure typical f ...

, events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of event ...

, equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, differential structureIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s, and categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...

.
Sometimes, a set is endowed with more than one structure 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 structure and a group structure, such that these two structures are related in a certain way, then the set becomes a topological group
350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition
In mathematics, a topological group is a group ...

.
Mappings between sets which preserve structures (i.e., structures in the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

are mapped to equivalent structures in the codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

) are of special interest in many fields of mathematics. Examples are homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

s, which preserve algebraic structures; homeomorphism
and a donut (torus
In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle.
If the axis of ...

s, which preserve topological structures; and diffeomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, which preserve differential structures.
History

In 1939, the French group with the pseudonymNicolas 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 intended ...

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 ofreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s 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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

.
*A measure: intervals of the real line have a specific length
Length is a measure of distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

, which can be extended to the Lebesgue measureIn measure theory
In mathematics, a measure on a set (mathematics), set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization ...

on many of its subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s.
*A metric: there is a notion of distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

between points.
*A geometry: it is equipped with a metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

and is flat.
*A topology: there is a notion of open set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

.
*Its algebraic structure and topology make it into a Lie group
In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...

, a type of topological group
350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition
In mathematics, a topological group is a group ...

.
See also

* Abstract structure *Isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

* Equivalent definitions of mathematical structures
*Intuitionistic type theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory
In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meanin ...

*Space (mathematics)
In mathematics, a space is a Set (mathematics), set (sometimes called a Universe (mathematics), universe) with some added mathematical structure, structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear sp ...

References

Further reading

* * * * * *External links

* ''(provides a model theoretic definition.)''Mathematical structures in computer science

(journal) {{Authority control Type theory Set theory