Definition:Closure Operator/Ordering

Definition

Definition 1

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 $\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({x}\right)}\right)$ $\displaystyle =$ $\displaystyle \operatorname{cl} \left({x}\right)$

Definition 2

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 condition for all elements $x, y \in S$:

$x \preceq \operatorname{cl} \left({y}\right) \iff \operatorname{cl} \left({x}\right) \preceq \operatorname{cl} \left({y}\right)$