# Definition:Closure Operator

## Definition

### Ordering

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

### Power Set

When the ordering in question is the subset relation on a power set, the definition can be expressed as follows:

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

A closure operator on $S$ is a mapping:

$\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$

which satisfies the following conditions for all sets $X, Y \subseteq S$:

 $(1)$ $:$ $\operatorname{cl}$ is inflationary $\displaystyle \forall X \subseteq S:$ $\displaystyle X \subseteq \operatorname{cl} \left({X}\right)$ $(2)$ $:$ $\operatorname{cl}$ is increasing $\displaystyle \forall X, Y \subseteq S:$ $\displaystyle X \subseteq Y \implies \operatorname{cl} \left({X}\right) \subseteq \operatorname{cl} \left({Y}\right)$ $(3)$ $:$ $\operatorname{cl}$ is idempotent $\displaystyle \forall X \subseteq S:$ $\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right) = \operatorname{cl} \left({X}\right)$