Definition:Closure Operator

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


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


Also see