Definition:Closure Operator/Ordering

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 $\ds x$ $\ds \preceq$ $\ds \map \cl x$ $\cl$ is increasing $\ds x \preceq y$ $\ds \implies$ $\ds \map \cl x \preceq \map \cl y$ $\cl$ is idempotent $\ds \map \cl {\map \cl x}$ $\ds =$ $\ds \map \cl x$

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.