# 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 **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 9 subcategories, out of 9 total.

## Pages in category "Closure Operators"

The following 29 pages are in this category, out of 29 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