## Table of Contents

73 relations: Algebra of sets, Algebraic closure, Algebraic structure, Algebraic variety, Arity, Associative property, Axiom, Binary operation, Binary relation, Closure (topology), Closure operator, Commutative ring, Convex hull, Convex set, Countable set, Cyclic group, Equivalence relation, Existential quantification, Field (mathematics), Floor and ceiling functions, Formal language, Function (mathematics), Generator (mathematics), Geometry, Group (mathematics), Group theory, Ideal (ring theory), Idempotence, Identity (mathematics), Identity element, If and only if, Integral domain, Integral element, Intersection (set theory), Inverse element, Σ-algebra, Kleene star, Kuratowski closure axioms, Least-upper-bound property, Limit of a sequence, Linear algebra, Linear combination, Linear span, Mathematical analysis, Matroid, Monotonic function, Natural number, Normal closure (group theory), Operation (mathematics), Ordered pair, ... Expand index (23 more) »

- Closure operators

## Algebra of sets

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.

See Closure (mathematics) and Algebra of sets

## Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

See Closure (mathematics) and Algebraic closure

## Algebraic structure

In mathematics, an algebraic structure 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 multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. Closure (mathematics) and algebraic structure are abstract algebra.

See Closure (mathematics) and Algebraic structure

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.

See Closure (mathematics) and Algebraic variety

## Arity

In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation. Closure (mathematics) and arity are abstract algebra.

See Closure (mathematics) and Arity

## Associative property

In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result.

See Closure (mathematics) and Associative property

## Axiom

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

See Closure (mathematics) and Axiom

## Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element.

See Closure (mathematics) and Binary operation

## Binary relation

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain.

See Closure (mathematics) and Binary relation

## Closure (topology)

In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of. Closure (mathematics) and closure (topology) are closure operators.

See Closure (mathematics) and Closure (topology)

## Closure operator

In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself that satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive), |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing), |- | \operatorname(\operatorname(X)). Closure (mathematics) and closure operator are closure operators.

See Closure (mathematics) and Closure operator

## Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative.

See Closure (mathematics) and Commutative ring

## Convex hull

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. Closure (mathematics) and convex hull are closure operators.

See Closure (mathematics) and Convex hull

## Convex set

In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them.

See Closure (mathematics) and Convex set

## Countable set

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.

See Closure (mathematics) and Countable set

## Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

See Closure (mathematics) and Cyclic group

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

See Closure (mathematics) and Equivalence relation

## Existential quantification

In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

See Closure (mathematics) and Existential quantification

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. Closure (mathematics) and field (mathematics) are abstract algebra.

See Closure (mathematics) and Field (mathematics)

## Floor and ceiling functions

In mathematics, the floor function is the function that takes as input a real number, and gives as output the greatest integer less than or equal to, denoted or.

See Closure (mathematics) and Floor and ceiling functions

## Formal language

In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules called a formal grammar.

See Closure (mathematics) and Formal language

## Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Closure (mathematics) and Function (mathematics)

## Generator (mathematics)

In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. Closure (mathematics) and generator (mathematics) are abstract algebra.

See Closure (mathematics) and Generator (mathematics)

## Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

See Closure (mathematics) and Geometry

## Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Closure (mathematics) and Group (mathematics)

## Group theory

In abstract algebra, group theory studies the algebraic structures known as groups.

See Closure (mathematics) and Group theory

## Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements.

See Closure (mathematics) and Ideal (ring theory)

## Idempotence

Idempotence is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. Closure (mathematics) and Idempotence are closure operators.

See Closure (mathematics) and Idempotence

## Identity (mathematics)

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity.

See Closure (mathematics) and Identity (mathematics)

## Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied.

See Closure (mathematics) and Identity element

## If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

See Closure (mathematics) and If and only if

## Integral domain

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

See Closure (mathematics) and Integral domain

## Integral element

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

See Closure (mathematics) and Integral element

## Intersection (set theory)

In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.

See Closure (mathematics) and Intersection (set theory)

## Inverse element

In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers. Closure (mathematics) and inverse element are abstract algebra.

See Closure (mathematics) and Inverse element

## Σ-algebra

In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.

See Closure (mathematics) and Σ-algebra

## Kleene star

In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters.

See Closure (mathematics) and Kleene star

## Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. Closure (mathematics) and Kuratowski closure axioms are closure operators.

See Closure (mathematics) and Kuratowski closure axioms

## Least-upper-bound property

In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers.

See Closure (mathematics) and Least-upper-bound property

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1.

See Closure (mathematics) and Limit of a sequence

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices.

See Closure (mathematics) and Linear algebra

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

See Closure (mathematics) and Linear combination

## Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted, pp. Closure (mathematics) and linear span are abstract algebra.

See Closure (mathematics) and Linear span

## Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

See Closure (mathematics) and Mathematical analysis

## Matroid

In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. Closure (mathematics) and matroid are closure operators.

See Closure (mathematics) and Matroid

## Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

See Closure (mathematics) and Monotonic function

## Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

See Closure (mathematics) and Natural number

## Normal closure (group theory)

In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S. Closure (mathematics) and normal closure (group theory) are closure operators.

See Closure (mathematics) and Normal closure (group theory)

## Operation (mathematics)

In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value.

See Closure (mathematics) and Operation (mathematics)

## Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

See Closure (mathematics) and Ordered pair

## Partial function

In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to.

See Closure (mathematics) and Partial function

## Partially ordered set

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.

See Closure (mathematics) and Partially ordered set

## Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

See Closure (mathematics) and Polynomial

## Preorder

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.

See Closure (mathematics) and Preorder

## Principal ideal

In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P.

See Closure (mathematics) and Principal ideal

## Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability.

See Closure (mathematics) and Probability theory

## Radical of an ideal

In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called radicalization. Closure (mathematics) and radical of an ideal are closure operators.

See Closure (mathematics) and Radical of an ideal

## Reflexive closure

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R. A relation is called if it relates every element of X to itself. Closure (mathematics) and reflexive closure are closure operators.

See Closure (mathematics) and Reflexive closure

## Reflexive relation

In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself.

See Closure (mathematics) and Reflexive relation

## Set (mathematics)

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. Closure (mathematics) and set (mathematics) are set theory.

See Closure (mathematics) and Set (mathematics)

## Subgroup

In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.

See Closure (mathematics) and Subgroup

## Subset

In mathematics, a set A is a subset of a set B if all 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 are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).

See Closure (mathematics) and Subset

## Substructure (mathematics)

In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain.

See Closure (mathematics) and Substructure (mathematics)

## Symmetric closure

In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X that contains R. For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y", then the symmetric closure of R is the relation "there is a direct flight either from x to y or from y to x". Closure (mathematics) and symmetric closure are closure operators.

See Closure (mathematics) and Symmetric closure

## Symmetric relation

A symmetric relation is a type of binary relation.

See Closure (mathematics) and Symmetric relation

## Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

See Closure (mathematics) and Topological space

## Transitive closure

In mathematics, the transitive closure of a homogeneous binary relation on a set is the smallest relation on that contains and is transitive. Closure (mathematics) and transitive closure are closure operators.

See Closure (mathematics) and Transitive closure

## Transitive relation

In mathematics, a binary relation on a set is transitive if, for all elements,, in, whenever relates to and to, then also relates to.

See Closure (mathematics) and Transitive relation

## Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. a single input.

See Closure (mathematics) and Unary operation

## Universal quantification

In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any".

See Closure (mathematics) and Universal quantification

## Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

See Closure (mathematics) and Vector space

## Zariski topology

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties.

See Closure (mathematics) and Zariski topology

## Zero of a function

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) vanishes at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x).

See Closure (mathematics) and Zero of a function

## See also

### Closure operators

- Affine hull
- Alexandrov topology
- Approach space
- Closure (mathematics)
- Closure (topology)
- Closure operator
- Complete lattice
- Convex hull
- Fixed-point theorem
- Galois connection
- Idempotence
- Interior (topology)
- Interior algebra
- Kuratowski closure axioms
- Lawvere–Tierney topology
- Matroid
- Matroid theory
- Monadic Boolean algebra
- Normal closure (group theory)
- Preclosure operator
- Proximity space
- Radical of an ideal
- Reflexive closure
- Symmetric closure
- Transitive closure

## References

Also known as Abstract closure, Abstract closure operator, Additively closed, Axiom of closure, Closed under, Closure (binary operation), Closure of a relation, Closure property, Closure property of multiplication, Congruence closure, Equivalence closure, P closure, P closure (binary relation), Reflexive symmetric transitive closure, Reflexive transitive closure, Reflexive transitive symmetric closure, Set closure (mathematics).