closure map

Let P be a poset. A function c:PP is called a closure map if

  • c is order preserving,

  • 1Pc,

  • c is idempotent: cc=c.

If the second condition is changed to c1P, then c is called a dual closure map on P.

For example, the real function f such that f(r) is the least integer greater than or equal to r is a closure map (see Archimedean property). The rounding function [] is an example of a dual closure map.

A fixed point of a closure map c on P is an element xP such that c(x)=x. It is evident that every image point of c is a fixed point: for if x=c(a) for some aP, then c(x)=c(c(a))=c(a)=x.

In the example above, any integer is a fixed point of f.

Every closure map can be characterized by an interesting decomposition property: c:PP is a closure map iff there is a set Q and a residuated function f:PQ such that c=f+f, where f+ denotes the residual of f.

Again, in the example above, f=g+g, where g: is the function taking any real number r to the largest integer smaller than r. g is residuated, and its residual is g+(x)=x+1.

Remark. Closure maps are generalizationsPlanetmathPlanetmath to closure operator on a set (see the parent entry). Indeed, any closure operator on a set X takes a subset A of X to a subset Ac of X satisfying the closure axioms, where Axiom 2 corresponds to condition 2 above, and Axiom 3 says the operator is idempotent. To see that the operator is order preserving, suppose AB. Then Bc=(AB)c=AcBc by Axiom 4, and hence AcBc. Axiom 1 says that the empty setMathworldPlanetmath is a fixed point of the operator. However, in general, this is not the case, for P may not even have a minimal element, as indicated by the above example.


Title closure map
Canonical name ClosureMap
Date of creation 2013-03-22 18:53:55
Last modified on 2013-03-22 18:53:55
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 54A05
Classification msc 06A15
Synonym closure
Synonym closure function
Synonym closure operator
Defines dual closure
Defines fixed point