# Definition:Closure Operator

## Definition

### Ordering

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 $\displaystyle \map \cl {\map \cl x}$ $\displaystyle =$ $\displaystyle \map \cl x$

### 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.

Let $\powerset S$ denote the power set of $S$.

A closure operator on $S$ is a mapping:

$\cl: \powerset S \to \powerset S$

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

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