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 closure axioms as follows for all elements $x, y \in S$:

 $(\text {cl} 1)$ $:$ $\cl$ is inflationary: $\ds x$ $\ds \preceq$ $\ds \map \cl x$ $(\text {cl} 2)$ $:$ $\cl$ is increasing: $\ds x \preceq y$ $\ds \implies$ $\ds \map \cl x \preceq \map \cl y$ $(\text {cl} 3)$ $:$ $\cl$ 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 closure axioms as follows for all sets $X, Y \subseteq S$:

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