Category:Definitions/Closure Operators

From ProofWiki
Jump to: navigation, search

This category contains definitions related to Closure Operators.
Related results can be found in Category:Closure Operators.


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