Category:Definitions/Examples of Closure Operators

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This category contains definitions of examples of Closure Operator.


Ordering

Let $\struct {S, \preceq}$ be an ordered set.


A closure operator on $S$ is a mapping:

$\cl: S \to S$

which satisfies the closure axioms as follows for all elements $x, y \in S$:

\((\text {cl} 1)\)   $:$   $\cl$ is inflationary:       \(\ds x \)   \(\ds \preceq \)   \(\ds \map \cl x \)      
\((\text {cl} 2)\)   $:$   $\cl$ is increasing:       \(\ds x \preceq y \)   \(\ds \implies \)   \(\ds \map \cl x \preceq \map \cl y \)      
\((\text {cl} 3)\)   $:$   $\cl$ is idempotent:       \(\ds \map \cl {\map \cl x} \)   \(\ds = \)   \(\ds \map \cl x \)      


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 closure axioms as follows for all sets $X, Y \subseteq S$:

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

Subcategories

This category has only the following subcategory.

Pages in category "Definitions/Examples of Closure Operators"

The following 2 pages are in this category, out of 2 total.