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 $\varnothing$ 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.

Bibliography

1

T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).