# Definition:Closure Operator

## 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.