# 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 $\left({S, \preceq}\right)$ 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 | \(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({x}\right)}\right) \) | \(\displaystyle = \) | \(\displaystyle \operatorname{cl} \left({x}\right) \) |

## Subcategories

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

### E

### L

### R

### S

### T

### U

## Pages in category "Closure Operators"

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

### C

- Closed Element of Composite Closure Operator
- Closed Elements Uniquely Determine Closure Operator
- Closure is Closed
- Closure is Closed/Power Set
- Closure is Smallest Closed Successor
- Closure Operator does not Change Infimum of Subset of Image
- Closure Operator from Closed Elements
- Closure Operator from Closed Sets
- Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema
- Composition of Compatible Closure Operators
- Compositions of Closure Operators are both Closure Operators iff Operators Commute

### E

### I

### O

- Operator Generated by Closure System is Closure Operator
- Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema
- Operator Generated by Image of Closure Operator is Closure Operator
- Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic
- Ordering on Closure Operators iff Images are Including