closure map
Let $P$ be a poset. A function $c:P\to P$ is called a closure map if

•
$c$ is order preserving,

•
${1}_{P}\le c$,

•
$c$ is idempotent: $c\circ c=c$.
If the second condition is changed to $c\le {1}_{P}$, 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 $[\cdot ]$ is an example of a dual closure map.
A fixed point of a closure map $c$ on $P$ is an element $x\in P$ 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 $a\in P$, 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: $c:P\to P$ is a closure map iff there is a set $Q$ and a residuated function $f:P\to Q$ such that $c={f}^{+}\circ f$, where ${f}^{+}$ denotes the residual of $f$.
Again, in the example above, $f={g}^{+}\circ g$, where $g:\mathbb{R}\to \mathbb{Z}$ 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 ${A}^{c}$ 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 $A\subseteq B$. Then ${B}^{c}={(A\cup B)}^{c}={A}^{c}\cup {B}^{c}$ by Axiom 4, and hence ${A}^{c}\subseteq {B}^{c}$. Axiom 1 says that the empty set^{} $\mathrm{\varnothing}$ 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  20130322 18:53:55 
Last modified on  20130322 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 