Definition:Closure Operator/Ordering
< Definition:Closure Operator(Redirected from Definition:Closure Operator (Order Theory))
Definition
Definition 1
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 | \(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({x}\right)}\right) \) | \(\displaystyle = \) | \(\displaystyle \operatorname{cl} \left({x}\right) \) |
Definition 2
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 condition for all elements $x, y \in S$:
- $x \preceq \operatorname{cl} \left({y}\right) \iff \operatorname{cl} \left({x}\right) \preceq \operatorname{cl} \left({y}\right)$
Also see
- Equivalence of Definitions of Closure Operator
- Results about closure operators can be found here.