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


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.