# 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.