Mathematical structure in the context of Field of sets

Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. In other words, to be abstract is to remove context and application. Two of the most highly abstract areas of modern mathematics are category theory and model theory.

A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.

In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set.A subspace is a subset of the parent space which retains the same structure.While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.

A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spaces are considered identical.

In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects y_i conform, in some consistent way, to those existing among the corresponding represented objects x_i. More specifically, given a set Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphism that preserves Π. The label representation is sometimes also applied to the homomorphism itself (such as group homomorphism in group theory).

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Many areas of computer science also rely on category theory, such as functional programming and semantics.

In mathematics, a group action of a group ${\displaystyle G}$ on a set ${\displaystyle S}$ is a group homomorphism from ${\displaystyle G}$ to some group (under function composition) of functions from ${\displaystyle S}$ to itself. It is said that ${\displaystyle G}$ acts on ${\displaystyle S}$.

Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.

In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as ⁠${\displaystyle A\cong B}$⁠. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified.

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. Often in mathematical jargon, especially in analysis or geometry, a function could refer to a map of the form ${\displaystyle X\to \mathbb {R} }$ or ${\displaystyle X\to \mathbb {C} }$ where ${\displaystyle X}$ is the space in question. Whilst other maps of the form ${\displaystyle X\to Y}$ between any two spaces are simply referred to as maps. Example of this can be the space of compactly supported functions on a topological space. However in a larger context a function space could just consist of a set of functions (set theoretically) equipped with possibly some extra structure.

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.

Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

When some object ${\displaystyle X}$ is said to be embedded in another object ${\displaystyle Y}$, the embedding is given by some injective and structure-preserving map ${\displaystyle f:X\rightarrow Y}$. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ${\displaystyle X}$ and ${\displaystyle Y}$ are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.

Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being ⁠${\displaystyle \varnothing }$⁠ and the top being the universe set under consideration.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called primal). Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".

In mathematics, an action of a group ${\displaystyle G}$ on a set ${\displaystyle S}$ is, loosely speaking, an operation that takes an element of ${\displaystyle G}$ and an element of ${\displaystyle S}$ and produces another element of ${\displaystyle S.}$ More formally, it is a group homomorphism from ${\displaystyle G}$ to the automorphism group of X (the set of all bijections on ${\displaystyle X}$ along with group operation being function composition). One says that ${\displaystyle G}$ acts on ${\displaystyle S.}$

Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).

Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.