# Closure operator

In mathematics, a **closure operator** on a set *S* is a function from the power set of *S* to itself that satisfies the following conditions for all sets

(cl is *extensive*),(cl is *monotone*),(cl is *idempotent*).

Closure operators are determined by their **closed sets**, i.e., by the sets of the form cl(*X*), since the **closure** cl(*X*) of a set *X* is the smallest closed set containing *X*. Such families of "closed sets" are sometimes called **closure systems** or "**Moore families**", in honor of E. H. Moore who studied closure operators in his 1910 *Introduction to a form of general analysis*, whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces.^{[1]} Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind
and George Cantor.^{[2]}

Closure operators are also called "**hull operators**", which prevents confusion with the "closure operators" studied in topology. A set together with a closure operator on it is sometimes called a **closure space**.

## ApplicationsEdit

Closure operators have many applications:

In topology, the closure operators are *topological* closure operators, which must satisfy

for all (Note that for this gives ).

In algebra and logic, many closure operators are **finitary closure operators**, i.e. they satisfy

In the theory of partially ordered sets, which are important in theoretical computer science, closure operators have a more general definition that replaces with . (See § Closure operators on partially ordered sets.)

## Closure operators in topologyEdit

The topological closure of a subset *X* of a topological space consists of all points *y* of the space, such that every neighbourhood of *y* contains a point of *X*. The function that associates to every subset *X* 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.

## Closure operators in algebraEdit

Finitary closure operators play a relatively prominent role in universal algebra, and in this context they are traditionally called *algebraic closure operators*. Every subset of an algebra *generates* a subalgebra: 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 given vector space its linear span. Similarly, the function that associates to every subset of a given group the subgroup generated by it, and similarly for fields and all other types of algebraic structures.

The linear span in a vector space and the similar algebraic closure in a field both satisfy the *exchange property:* If *x* is in the closure of the union of *A* and {*y*} but not in the closure of *A*, then *y* is in the closure of the union of *A* and {*x*}. A finitary closure operator with this property is called a matroid. The dimension of a vector space, or the transcendence degree of a field (over its prime field) is exactly the rank of the corresponding matroid.

The function that maps every subset of a given field to its algebraic 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 in model theory as dcl (for *definable closure*) and acl (for *algebraic closure*).

The convex hull in *n*-dimensional Euclidean space is another example of a finitary closure operator. It satisfies the *anti-exchange property:* If *x* is in the closure of the union of {*y*} and *A*, but not in the union of {*y*} and closure of *A*, then *y* is not in the closure of the union of {*x*} and *A*. Finitary closure operators with this property give rise to antimatroids.

As another example of a closure operator used in algebra, if some algebra has universe *A* and *X* is a set of pairs of *A*, then the operator assigning to *X* the smallest congruence containing *X* is a finitary closure operator on *A x A*.^{[3]}

## Closure operators in logicEdit

Suppose you have some logical formalism that contains certain rules allowing you to derive new formulas from given ones. Consider the set *F* of all possible formulas, and let *P* be the power set of *F*, ordered by ⊆. For a set *X* of formulas, let cl(*X*) be the set of all formulas that can be derived from *X*. Then cl is a closure operator on *P*. More precisely, we can obtain cl as follows. Call "continuous" an operator *J* such that, for every directed class *T*,

*J*(*lim T*)=*lim J*(*T*).

This continuity condition is on the basis of a fixed point theorem for *J*. Consider the one-step operator *J* of a monotone logic. This is the operator associating any set *X* of formulas with the set *J*(*X*) of formulas that are either logical axioms or are obtained by an inference rule from formulas in *X* or are in *X*. Then such an operator is continuous and we can define cl(*X*) as the least fixed point for *J* greater or equal to *X*. 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 in fuzzy logic (see Gerla 2000).

### Consequence operatorsEdit

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 of *sentences*). In abstract algebraic logic, finitary closure operators are still studied under the name *consequence operator*, which was coined by Tarski. The set *S* represents a set of sentences, a subset *T* of *S* 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 called **finite consequence operators**.

## Closed and pseudo-closed setsEdit

The closed sets with respect to a closure operator on *S* form a subset *C* of the power set * P*(

*S*). Any intersection of sets in

*C*is again in

*C*. In other words,

*C*is a complete meet-subsemilattice of

*(*

**P***S*). Conversely, if

*C*⊆

*(*

**P***S*) is closed under arbitrary intersections, then the function that associates to every subset

*X*of

*S*the smallest set

*Y*∈

*C*such that

*X*⊆

*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-complete sublattice of * P*(

*S*). Even for non-topological closure operators,

*C*can be seen as having the structure of a lattice. (The join of two sets

*X*,

*Y*⊆

*(*

**P***S*) being cl(

*X*

*Y*).) But then

*C*is not a sublattice of the lattice

*(*

**P***S*).

Given a finitary closure operator on a set, the closures of finite sets are exactly the compact elements of the set *C* of closed sets. It follows that *C* is an algebraic poset.
Since *C* is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, if *C* is an algebraic poset, then the closure operator is finitary.

Each closure operator on a finite set *S* is uniquely determined by its images of its *pseudo-closed* sets.^{[5]}
These are recursively defined: A set is **pseudo-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- if
*Q*⊂*P*is pseudo-closed, then cl(*Q*) ⊆*P*.

## Closure operators on partially ordered setsEdit

A partially ordered set (poset) is a set together with a *partial order* ≤, i.e. a binary relation that is reflexive (*a* ≤ *a*), transitive (*a* ≤ *b* ≤ *c* implies *a* ≤ *c*) and antisymmetric (*a* ≤ *b* ≤ *a* implies *a* = *b*). Every power set **P**(*S*) together with inclusion ⊆ is a partially ordered set.

A function cl: *P* → *P* from a partial order *P* to itself is called a closure operator if it satisfies the following axioms for all elements *x*, *y* in *P*.

*x*≤ cl(*x*)(cl is *extensive*)*x*≤*y*implies cl(*x*) ≤ cl(*y*)(cl is increasing) cl(cl( *x*)) = cl(*x*)(cl is idempotent)

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 all *x*, *y* in *P*.

Using the pointwise order on functions between posets, one may alternatively write the extensiveness property as id_{P} ≤ cl, where id is the identity function. A self-map *k* that is increasing and idempotent, but satisfies the dual of the extensiveness property, i.e. *k* ≤ id_{P} is called a **kernel operator**,^{[6]} **interior operator**,^{[7]} or **dual closure**.^{[8]} As examples, if *A* is a subset of a set *B*, then the self-map on the powerset of *B* given by *μ _{A}*(

*X*) =

*A*∪

*X*is a closure operator, whereas

*λ*(

_{A}*X*) =

*A*∩

*X*is a kernel operator. The ceiling function from the real numbers to the real numbers, which assigns to every real

*x*the smallest integer not smaller than

*x*, is another example of a closure operator.

A fixpoint of the function cl, i.e. an element *c* of *P* that satisfies cl(*c*) = *c*, is called a **closed element**. A closure operator on a partially ordered set is determined by its closed elements. If *c* is a closed element, then *x* ≤ *c* and cl(*x*) ≤ *c* are equivalent conditions.

Every Galois connection (or residuated 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.^{[9]} 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: if *A* is the set of closed elements with respect to cl, then cl: *P* → *A* is the lower adjoint of a Galois connection between *P* and *A*, with the upper adjoint being the embedding of *A* into *P*. Furthermore, every lower adjoint of an embedding of some subset into *P* is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint.

Any partially ordered set *P* can be viewed as a category, with a single morphism from *x* to *y* if and only if *x* ≤ *y*. The closure operators on the partially ordered set *P* are then nothing but the monads on the category *P*. Equivalently, a closure operator can be viewed as an endofunctor on the category of partially ordered sets that has the additional *idempotent* and *extensive* properties.

If *P* is a complete lattice, then a subset *A* of *P* is the set of closed elements for some closure operator on *P* if and only if *A* is a **Moore family** on *P*, i.e. the largest element of *P* is in *A*, and the infimum (meet) of any non-empty subset of *A* is again in *A*. Any such set *A* is itself a complete lattice with the order inherited from *P* (but the supremum (join) operation might differ from that of *P*). When *P* is the powerset Boolean algebra of a set *X*, then a Moore family on *P* is called a **closure system** on *X*.

The closure operators on *P* form themselves a complete lattice; the order on closure operators is defined by cl_{1} ≤ cl_{2} iff cl_{1}(*x*) ≤ cl_{2}(*x*) for all *x* in *P*.

## See alsoEdit

## NotesEdit

**^**Blyth p.11**^**Marcel Erné,*Closure*, in Frédéric Mynard, Elliott Pearl (Editors),*Beyond Topology*, Contemporary mathematics vol. 486, American Mathematical Society, 2009.**^**Clifford Bergman,*Universal Algebra*, 2012, Section 2.4.**^**Ganter, Algorithm 1**^**Ganter, Section 3.2**^**Giertz, p. 26**^**Erné, p. 2, uses closure (resp. interior) operation**^**Blyth, p. 10**^**Blyth, p. 10

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## External linksEdit

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