Definition:Closure Operator

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This page is about Closure Operator. For other uses, see Closure.

Definition

Ordering

Let $\struct {S, \preceq}$ be an ordered set.


A closure operator on $S$ is a mapping:

$\cl: S \to S$

which satisfies the following conditions for all elements $x, y \in S$:

$\cl$ is inflationary       \(\ds x \)   \(\ds \preceq \)   \(\ds \map \cl x \)             
$\cl$ is increasing       \(\ds x \preceq y \)   \(\ds \implies \)   \(\ds \map \cl x \preceq \map \cl y \)             
$\cl$ is idempotent       \(\ds \map \cl {\map \cl x} \)   \(\ds = \)   \(\ds \map \cl x \)             


Power Set

When the ordering in question is the subset relation on a power set, the definition can be expressed as follows:


Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.


A closure operator on $S$ is a mapping:

$\cl: \powerset S \to \powerset S$

which satisfies the following conditions for all sets $X, Y \subseteq S$:

\((1)\)   $:$   $\cl$ is inflationary      \(\ds \forall X \subseteq S:\)    \(\ds X \)   \(\ds \subseteq \)   \(\ds \map \cl X \)             
\((2)\)   $:$   $\cl$ is increasing      \(\ds \forall X, Y \subseteq S:\)    \(\ds X \subseteq Y \)   \(\ds \implies \)   \(\ds \map \cl X \subseteq \map \cl Y \)             
\((3)\)   $:$   $\cl$ is idempotent      \(\ds \forall X \subseteq S:\)    \(\ds \map \cl {\map \cl X} \)   \(\ds = \)   \(\ds \map \cl X \)             


Notation

The closure operator of $H$ is variously denoted:

$\map \cl H$
$\map {\mathrm {Cl} } H$
$\overline H$
$H^-$


Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement.

$\map \cl H$ and $\map {\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear.

$H^-$ is neat and compact, but has the disadvantage of being relatively obscure.


On $\mathsf{Pr} \infty \mathsf{fWiki}$, $H^-$ is notation of choice, although $\map \cl H$ can also be found in places.


Also see