Category:Definitions/Closure Operators
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This category contains definitions related to Closure Operators.
Related results can be found in Category: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 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: | \(\ds \map \cl {\map \cl x} \) | \(\ds = \) | \(\ds \map \cl x \) |
Subcategories
This category has the following 6 subcategories, out of 6 total.
C
- Definitions/Closed Elements (6 P)
E
L
- Definitions/Lower Sections (8 P)
R
S
U
- Definitions/Upper Sections (5 P)
Pages in category "Definitions/Closure Operators"
The following 21 pages are in this category, out of 21 total.
C
- Definition:Closed Element
- Definition:Closed Set/Closure Operator
- Definition:Closure of Set under Closure Operator
- Definition:Closure Operator
- Definition:Closure Operator (Order Theory)
- Definition:Closure Operator on Set
- Definition:Closure Operator/Ordering
- Definition:Closure Operator/Ordering/Definition 1
- Definition:Closure Operator/Ordering/Definition 2
- Definition:Closure Operator/Power Set