Closure operator

In mathematics we mean by the closure of a set is a superset that is large enough to meet certain requirements, and at the same time is the smallest set that satisfies these requirements. Examples are the convex hull of a subset of a vector space, the closure of a subset of a topological space or the transitive closure of a binary relation. Closure operator means the provision by which each set of objects is associated with its case.

Definitions

A casing operator is extensive, monotonous, idempotent illustration, which in turn associates each subset of a given set is a subset of, namely the shell. Specifically, the requirements mean:

Because the other two requirements, it is sufficient to demand, instead of only idempotent, ie: forming of the hull of a set again the shell, so nothing more is added.

Equivalent to the three aforementioned receivables is as follows. called closure operator if for all

A closure operator is also called completion operator, because to a structured set ( a topological space, an algebraic structure) belonging closure operator any subset of these structured amount to the smallest sub-structure reflects that contains this subset. However, the sub-structures ( closed sets in topological space, algebraic substructures ) just make with respect to the given structure completed subsets.

Shell systems

Definition

A containment system is a closed under arbitrary intersection of sets of sets, that is, a containment system on a set is one of subsets of existing amount with the following properties:

• Is an element of.
• For each non-empty subset of the intersection of the elements of an item, or otherwise expressed: The intersection of any number of elements is itself an element of.

Considering as the basic amount, so it makes sense in this context, the general set theory not defined average over the empty set as

Define, because only then is achieved. Thus, the above two conditions simplify to the single - equivalent - condition

• For each subset of the intersection of the elements of an element is off.

Relationship between envelope systems and shell operators

Shell operators and envelope systems correspond to each other:

• Is a cover system, then one can define a closure operator as follows:

• Conversely, a case can be obtained in case of any system operator:

In many applications, it has been only a monotonic operator, that is, for each subset, and also from the following. For example, for a topological space, each point set to assign the set of its accumulation points, or - if a semigroup is - lots of the set of all products of elements from. An associated closure operator is then obtained in one of two ways:

• Be the intersection of all supersets with
• Be the union of the sets

Both versions provide the same closure operator. This is called taking also the conclusion. The first option is usually cheaper for theoretical, the second practical applications.

Examples

• Consider the plane R2. The convex part of the amounts of plane form a barrier system, the associated sheath operator is the formation of the convex hull of a subset.
• The surrounding minimal rectangle is a closure operator for intervals (box ).
• The amounts of a topological space form a closed shell system. The associated closure operator causes the formation of the completed hull of a subset of the underlying topological space and is sometimes referred to by the Polish mathematician Kazimierz Kuratowski as Kuratowskischer closure operator. The closure of a subset of a topological space is the smallest superset of that is closed under limit formation of networks on the respective amount.
• If a group is given, then its subgroups form a containment system. The associated closure operator is the formation of the group that is generated by a subset.
• The normal subgroups of a group form a containment system.
• Every ideal system is an envelope system.
• The formation of the transitive closure of a relation is a closure operator.
• The two chains and a Galois connection are shell operators.
• The formation of the clover ash shell of a formal language is a closure operator.
• The σ - operator from the measure theory, which assigns to each set of subsets of a space is the smallest complete σ -algebra is a closure operator.
• The inference of formal logic is a closure operator.
• For the enveloping body to a set of numbers is required that for all elements of the set always their sum, their product, their difference and their quotient (except division by zero) and the numbers 1 and 0 belong to the set. The enveloping body of the set { 0} is therefore already the set of all rational numbers. Only when the number set at least one irrational number contains (for example, √ 2), the result is a body that comprises genuine.
• In each subcategory of Set, which contains as morphisms only inclusion illustrations, each monad is a closure operator.

Applications to Formal Languages ​​and Complexity Classes

It is a class of formal languages ​​. We consider the following shell operators on:

• : Completed in under homomorphisms:

• : Completed in under - free homomorphisms, such as, but

• : Completed in under inverse homomorphisms:

• : Completed in under union:

• : Completed in under average:

• : Completed in under concatenation:

• : Completed in under Kleene star:

Under the corresponding operation if a class and one of the above envelope operators has the property that is true, then is called closed ( homomorphism - free homomorphism, inverse homomorphism, union, intersection, concatenation and Kleene star).