# Category:Reflexive Closures

This category contains results about Reflexive Closures.

### Definition 1

Let $\mathcal R$ be a relation on a set $S$.

The **reflexive closure** of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as:

- $\mathcal R^= := \mathcal R \cup \set {\tuple {x, x}: x \in S}$

That is:

- $\mathcal R^= := \mathcal R \cup \Delta_S$

where $\Delta_S$ is the diagonal relation on $S$.

### Definition 2

Let $\RR$ be a relation on a set $S$.

The **reflexive closure** of $\RR$ is defined as the smallest reflexive relation on $S$ that contains $\RR$ as a subset.

The **reflexive closure** of $\RR$ is denoted $\RR^=$.

### Definition 3

Let $\mathcal R$ be a relation on a set $S$.

Let $\mathcal Q$ be the set of all reflexive relations on $S$ that contain $\mathcal R$.

The **reflexive closure** of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as:

- $\mathcal R^= := \bigcap \mathcal Q$

That is:

- $\mathcal R^=$ is the intersection of all reflexive relations on $S$ containing $\mathcal R$.

## Also see

## Subcategories

This category has only the following subcategory.

### R

## Pages in category "Reflexive Closures"

The following 14 pages are in this category, out of 14 total.

### E

### R

- Reflexive Closure is Closure Operator
- Reflexive Closure is Idempotent
- Reflexive Closure is Inflationary
- Reflexive Closure is Order Preserving
- Reflexive Closure is Reflexive
- Reflexive Closure of Antisymmetric Relation is Antisymmetric
- Reflexive Closure of Relation Compatible with Operation is Compatible
- Reflexive Closure of Strict Ordering is Ordering
- Reflexive Closure of Strict Total Ordering is Total Ordering
- Reflexive Closure of Transitive Antisymmetric Relation is Ordering
- Reflexive Closure of Transitive Relation is Transitive