Definition:Closure Operator/Ordering

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Definition

Definition 1

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       \(\displaystyle \map \cl {\map \cl x} \)   \(\displaystyle = \)   \(\displaystyle \map \cl x \)             


Definition 2

Let $\struct {S, \preceq}$ be an ordered set.


A closure operator on $S$ is a mapping:

$\cl: S \to S$

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

$x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$


Also see

  • Results about closure operators can be found here.