closure map
Let P be a poset. A function c:P→P is called a closure map if
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c is order preserving,
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1P≤c,
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c is idempotent: c∘c=c.
If the second condition is changed to c≤1P, 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 x∈P 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 a∈P, 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:P→P is a closure map iff there is a set Q and a residuated function f:P→Q such that c=f+∘f, where f+ denotes the residual of f.
Again, in the example above, f=g+∘g, where g:ℝ→ℤ 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 generalizations 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 A⊆B. Then Bc=(A∪B)c=Ac∪Bc by Axiom 4, and hence Ac⊆Bc. Axiom 1 says that the empty set ∅ 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.
References
- 1 T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).
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 |