Inmathematics, aclosure operator on asetS is afunction${\displaystyle \mathrm{cl}:\mathcal{P}(S)\to \mathcal{P}(S)}$ from thepower set ofS to itself that satisfies the following conditions for all sets${\displaystyle X,Y\subseteq S}$

${\displaystyle X\subseteq \mathrm{cl}(X)}$	(cl isextensive),
${\displaystyle X\subseteq Y\Rightarrow \mathrm{cl}(X)\subseteq \mathrm{cl}(Y)}$	(cl isincreasing),
${\displaystyle \mathrm{cl}(\mathrm{cl}(X))=\mathrm{cl}(X)}$	(cl isidempotent).

Closure operators are determined by theirclosed sets, i.e., by the sets of the form cl(X), since theclosure cl(X) of a setX is the smallest closed set containingX. Such families of "closed sets" are sometimes calledclosure systems or "Moore families".^[1] A set together with a closure operator on it is sometimes called aclosure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied intopology.

E. H. Moore studied closure operators in his 1910Introduction to a form of general analysis, whereas the concept of the closure of a subset originated in the work ofFrigyes Riesz in connection with topological spaces.^[2] Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions byErnst Schröder,Richard Dedekind andGeorg Cantor.^[3]

The closed sets with respect to a closure operator onS form a subsetC of the power setP(S). Any intersection of sets inC is again inC. In other words,C is a completemeet-subsemilattice ofP(S). Conversely, ifC ⊆P(S) is closed under arbitrary intersections, then the function that associates to every subsetX ofS the smallest setY ∈C such thatX ⊆Y is a closure operator.

There is a simple and fast algorithm for generating all closed sets of a given closure operator.^[4]

A closure operator on a set is topological if and only if the set of closed sets is closed under finite unions, i.e.,C is a meet-completesublattice ofP(S). Even for non-topological closure operators,C can be seen as having the structure of a lattice. (The join of two setsX,Y ⊆P(S) being cl(X${\displaystyle \cup }$Y).) But thenC is not a sublattice of the latticeP(S).

Given a finitary closure operator on a set, the closures of finite sets are exactly thecompact elements of the setC of closed sets. It follows thatC is analgebraic poset.SinceC is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, ifC is an algebraic poset, then the closure operator is finitary.

Each closure operator on a finite setS is uniquely determined by its images of itspseudo-closed sets.^[5]These are recursively defined: A set ispseudo-closed if it is not closed and contains the closure of each of its pseudo-closed proper subsets. Formally:P ⊆ S is pseudo-closed if and only if

• P ≠ cl(P) and
• ifQ ⊂ P is pseudo-closed, then cl(Q) ⊆ P.

The usual setclosure fromtopology is a closure operator. Other examples include thelinear span of a subset of avector space, theconvex hull oraffine hull of a subset of a vector space or thelower semicontinuous hull${\displaystyle \overline{f}}$ of a function${\displaystyle f:E\to \mathbb{R}\cup \{\pm \mathrm{\infty }\}}$, where${\displaystyle E}$ is e.g. anormed space, defined implicitly${\displaystyle \mathrm{epi}(\overline{f})=\overline{\mathrm{epi}(f)}}$, where${\displaystyle \mathrm{epi}(f)}$ is theepigraph of a function${\displaystyle f}$.

Therelative interior${\displaystyle \mathrm{ri}}$ is not a closure operator: although it is idempotent, it is not increasing and if${\displaystyle C_1}$ is a cube in${\displaystyle {\mathbb{R}}^3}$ and${\displaystyle C_2}$ is one of its faces, then${\displaystyle C_2\subset C_1}$, but${\displaystyle \mathrm{ri}(C_1)\ne \mathrm{\varnothing }\ne \mathrm{ri}(C_2)}$ and${\displaystyle \mathrm{ri}(C_1)\cap \mathrm{ri}(C_2)=\mathrm{\varnothing }}$, so it is not increasing.^[6]

In topology, the closure operators aretopological closure operators, which must satisfy

${\displaystyle \mathrm{cl}(X_1\cup \cdots \cup X_n)=\mathrm{cl}(X_1)\cup \cdots \cup \mathrm{cl}(X_n)}$

for all${\displaystyle n\in \mathbb{N}}$ (Note that for${\displaystyle n=0}$ this gives${\displaystyle \mathrm{cl}(\varnothing )=\varnothing }$).

Inalgebra andlogic, many closure operators arefinitary closure operators, i.e. they satisfy

${\displaystyle \mathrm{cl}(X)=\bigcup \left\{\mathrm{cl}(Y):Y\subseteq X\text{ and }Y\text{ finite}\right\}.}$

In the theory ofpartially ordered sets, which are important intheoretical computer science, closure operators have a more general definition that replaces${\displaystyle \subseteq }$ with${\displaystyle \le }$. (See§ Closure operators on partially ordered sets.)

Thetopological closure of a subsetX of atopological space consists of all pointsy of the space, such that everyneighbourhood ofy contains a point ofX. The function that associates to every subsetX its closure is a topological closure operator. Conversely, every topological closure operator on a set gives rise to a topological space whose closed sets are exactly the closed sets with respect to the closure operator.

Finitary closure operators play a relatively prominent role inuniversal algebra, and in this context they are traditionally calledalgebraic closure operators. Every subset of analgebragenerates asubalgebra: the smallest subalgebra containing the set. This gives rise to a finitary closure operator.

Perhaps the best known example for this is the function that associates to every subset of a givenvector space itslinear span. Similarly, the function that associates to every subset of a givengroup thesubgroup generated by it, and similarly forfields and all other types ofalgebraic structures.

The linear span in a vector space and the similar algebraic closure in a field both satisfy theexchange property: Ifx is in the closure of the union ofA and {y} but not in the closure ofA, theny is in the closure of the union ofA and {x}. A finitary closure operator with this property is called amatroid. Thedimension of a vector space, or thetranscendence degree of a field (over itsprime field) is exactly the rank of the corresponding matroid.

The function that maps every subset of a givenfield to itsalgebraic closure is also a finitary closure operator, and in general it is different from the operator mentioned before. Finitary closure operators that generalize these two operators are studied inmodel theory as dcl (fordefinable closure) and acl (foralgebraic closure).

Theconvex hull inn-dimensionalEuclidean space is another example of a finitary closure operator. It satisfies theanti-exchange property: Ifx is in the closure of the union of {y} andA, but not in the union of {y} and closure ofA, theny is not in the closure of the union of {x} andA. Finitary closure operators with this property give rise toantimatroids.

As another example of a closure operator used in algebra, if some algebra has universeA andX is a set of pairs ofA, then the operator assigning toX the smallestcongruence containingX is a finitary closure operator onA x A.^[7]

Suppose you have somelogical formalism that contains certain rules allowing you to derive new formulas from given ones. Consider the setF of all possible formulas, and letP be thepower set ofF, ordered by ⊆. For a setX of formulas, let cl(X) be the set of all formulas that can be derived fromX. Then cl is a closure operator onP. More precisely, we can obtain cl as follows. Call "continuous" an operatorJ such that, for everydirected classT,

J(limT) = limJ(T).

This continuity condition is on the basis of a fixed point theorem forJ. Consider the one-step operatorJ of a monotone logic. This is the operator associating any setX of formulas with the setJ(X) of formulas that are either logical axioms or are obtained by an inference rule from formulas inX or are inX. Then such an operator is continuous and we can define cl(X) as the least fixed point forJ greater or equal toX. In accordance with such a point of view, Tarski, Brown, Suszko and other authors proposed a general approach to logic based on closure operator theory. Also, such an idea is proposed in programming logic (see Lloyd 1987) and infuzzy logic (see Gerla 2000).

Around 1930,Alfred Tarski developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set ofsentences). Inabstract algebraic logic, finitary closure operators are still studied under the nameconsequence operator, which was coined by Tarski. The setS represents a set of sentences, a subsetT ofS a theory, and cl(T) is the set of all sentences that follow from the theory. Nowadays the term can refer to closure operators that need not be finitary; finitary closure operators are then sometimes calledfinite consequence operators.

Apartially ordered set (poset) is a set together with apartial order ≤, i.e. abinary relation that is reflexive (a ≤a), transitive (a ≤b ≤c impliesa ≤c) andantisymmetric (a ≤b ≤a impliesa = b). Everypower setP(S) together with inclusion ⊆ is a partially ordered set.

A function cl:P →P from a partial orderP to itself is called a closure operator if it satisfies the following axioms for all elementsx,y inP.

x ≤ cl(x)	(cl isextensive)
x ≤y implies cl(x) ≤ cl(y)	(cl isincreasing)
cl(cl(x)) = cl(x)	(cl isidempotent)

More succinct alternatives are available: the definition above is equivalent to the single axiom

x ≤ cl(y) if and only if cl(x) ≤ cl(y)

for allx,y inP.

Using thepointwise order on functions between posets, one may alternatively write the extensiveness property as id_P ≤ cl, where id is theidentity function. A self-mapk that is increasing and idempotent, but satisfies thedual of the extensiveness property, i.e.k ≤ id_P is called akernel operator,^[8]interior operator,^[9] ordual closure.^[10] As examples, ifA is a subset of a setB, then the self-map on the powerset ofB given byμ_A(X) =A ∪X is a closure operator, whereasλ_A(X) =A ∩X is a kernel operator. Theceiling function from thereal numbers to the real numbers, which assigns to every realx the smallestinteger not smaller thanx, is another example of a closure operator.

Afixpoint of the function cl, i.e. an elementc ofP that satisfies cl(c) = c, is called aclosed element. A closure operator on a partially ordered set is determined by its closed elements. Ifc is a closed element, thenx ≤c and cl(x) ≤c are equivalent conditions.

EveryGalois connection (orresiduated mapping) gives rise to a closure operator (as is explained in that article). In fact,every closure operator arises in this way from a suitable Galois connection.^[11] The Galois connection is not uniquely determined by the closure operator. One Galois connection that gives rise to the closure operator cl can be described as follows: ifA is the set of closed elements with respect to cl, then cl:P →A is the lower adjoint of a Galois connection betweenP andA, with the upper adjoint being the embedding ofA intoP. Furthermore, every lower adjoint of an embedding of some subset intoP is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint.

Any partially ordered setP can be viewed as acategory, with a single morphism fromx toy if and only ifx ≤y. The closure operators on the partially ordered setP are then nothing but themonads on the categoryP. Equivalently, a closure operator can be viewed as an endofunctor on the category of partially ordered sets that has the additionalidempotent andextensive properties.

IfP is acomplete lattice, then a subsetA ofP is the set of closed elements for some closure operator onP if and only ifA is aMoore family onP, i.e. the largest element ofP is inA, and theinfimum (meet) of any non-empty subset ofA is again inA. Any such setA is itself a complete lattice with the order inherited fromP (but thesupremum (join) operation might differ from that ofP). WhenP is thepowersetBoolean algebra of a setX, then a Moore family onP is called aclosure system onX.

The closure operators onP form themselves a complete lattice; the order on closure operators is defined by cl₁ ≤ cl₂iff cl₁(x) ≤ cl₂(x) for allx inP.

• Čech closure operator – Closure operatorPages displaying short descriptions of redirect targets
• Closure (topology) – All points and limit points in a subset of a topological space
• Galois connection – Particular correspondence between two partially ordered sets
• Interior algebra – Algebraic structure
• Interior (topology) – Largest open subset of some given set
• Kuratowski closure axioms – Axioms for defining a topology
• Preclosure operator – Closure operator

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• Stanford Encyclopedia of Philosophy: "Algebraic Propositional Logic"—by Ramon Jansana.

• Closure operators
• Universal algebra
• Order theory

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