Closure operator

In mathematics, a closure operator on a set S is a function

E. H. Moore 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.

[2] 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 Georg Cantor.

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

(See § Closure operators on partially ordered sets.)

Finitary closure operators play a relatively prominent role in universal algebra, and in this context they are traditionally called algebraic closure operators.

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.

A finitary closure operator with this property is called a 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.

The convex hull in n-dimensional Euclidean space is another example of a finitary closure operator.

[5] Suppose you have some logical formalism that contains certain rules allowing you to derive new formulas from given ones.

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.

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.

Conversely, if C is an algebraic poset, then the closure operator is finitary.

Formally: P ⊆ S is pseudo-closed if and only if 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).

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. More succinct alternatives are available: the definition above is equivalent to the single axiom for all x, y in P. Using the pointwise order on functions between posets, one may alternatively write the extensiveness property as idP ≤ 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 ≤ idP is called a kernel operator,[8] interior operator,[9] or dual closure.

A closure operator on a partially ordered set is determined by its closed elements.

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.

[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: 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.

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.