Definition:Closure Operator
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 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.
Also see
- Definition:Closed Set under Closure Operator
- Definition:Closed Element under Closure Operator
- Definition:Closure of Set under Closure Operator
- Definition:Closure of Element under Closure Operator
 | Work In Progress In particular: In progress: refactoring the below page and crafting the last remaining needed links to allow the below to be transcluded into this page a another specific instance of a closure operator on Power Set - which makes this most general concept of a closure operator so much neater. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
- Closure (Topology), which is demonstrated to be an instance of a closure operator in Topological Closure is Closure Operator.