Equivalence of Definitions of Closure Operator

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

Let $\cl: S \to S$ be a mapping.

Definition 1 implies Definition 2
Let $\cl$ be an inflationary, increasing and idempotent mapping.

It is necessary to show that for all $x, y \in S$:
 * $x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$

Sufficient Condition
Suppose that $\map \cl x \preceq \map \cl y$.

Definition 2 implies Definition 1
Suppose that:
 * $x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$

It is necessary to show that $\cl$ is inflationary, increasing and idempotent.

Inflationary
Let $x \in S$.

That is, $\cl$ is inflationary.

Increasing
It has been demonstrated that $\preceq$ is inflationary.

Let $x, y \in S$ such that $x \preceq y$.

Then:

That is, $\cl$ is increasing.

Idempotent
It has been demonstrated that $\preceq$ is inflationary.

Let $x \in S$.

Then:

Then as $\cl$ is inflationary:
 * $\map \cl x \preceq \map \cl {\map \cl x}$

As $\preceq$ is antisymmetric by dint of being an ordering:
 * $\map \cl {\map \cl x} = \map \cl x$

That is, $\cl$ is idempotent.

Thus $\cl$ has been shown to be inflationary, increasing and idempotent as required.