Definition:Closure Operator

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Definition

Ordering

Let $\left({S, \preceq}\right)$ be an ordered set.


A closure operator on $S$ is a mapping:

$\operatorname{cl}: S \to S$

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

$\operatorname{cl}$ is inflationary       \(\displaystyle x \)   \(\displaystyle \preceq \)   \(\displaystyle \operatorname{cl} \left({x}\right) \)             
$\operatorname{cl}$ is increasing       \(\displaystyle x \preceq y \)   \(\displaystyle \implies \)   \(\displaystyle \operatorname{cl} \left({x}\right) \preceq \operatorname{cl} \left({y}\right) \)             
$\operatorname{cl}$ is idempotent       \(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({x}\right)}\right) \)   \(\displaystyle = \)   \(\displaystyle \operatorname{cl} \left({x}\right) \)             


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 and let $\mathcal P \left({S}\right)$ be the power set of $S$.


A closure operator on $S$ is a mapping:

$\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$

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

\((1)\)   $:$   $\operatorname{cl}$ is inflationary      \(\displaystyle \forall X \subseteq S:\) \(\displaystyle X \subseteq \operatorname{cl} \left({X}\right) \)             
\((2)\)   $:$   $\operatorname{cl}$ is increasing      \(\displaystyle \forall X, Y \subseteq S:\) \(\displaystyle X \subseteq Y \implies \operatorname{cl} \left({X}\right) \subseteq \operatorname{cl} \left({Y}\right) \)             
\((3)\)   $:$   $\operatorname{cl}$ is idempotent      \(\displaystyle \forall X \subseteq S:\) \(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right) = \operatorname{cl} \left({X}\right) \)             


Also see