Definition:Closure Operator/Ordering/Definition 2

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Definition

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 $x, y \in S$:

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