Category:Closure Operators

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This category contains results about Closure Operators in the context of Mapping Theory.
Definitions specific to this category can be found in Definitions/Closure Operators.

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       \(\displaystyle \map \cl {\map \cl x} \)   \(\displaystyle = \)   \(\displaystyle \map \cl x \)             


This category has the following 7 subcategories, out of 7 total.





Pages in category "Closure Operators"

The following 30 pages are in this category, out of 30 total.