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.
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Pages in category "Definitions/Examples of Closure Operators"
The following 2 pages are in this category, out of 2 total.