# Category:Transitive Closures

This category contains results about **Transitive Closures** in the context of **Relation Theory**.

Definitions specific to this category can be found in **Definitions/Transitive Closures**.

### Smallest Transitive Superset

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

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

### Intersection of Transitive Supersets

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

The **transitive closure** of $\RR$ is defined as the intersection of all transitive relations on $S$ which contain $\RR$.

### Finite Chain

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

The **transitive closure** of $\RR$ is the relation $\RR^+$ defined as follows:

For $x, y \in S$, $x \mathrel {\RR^+} y$ if and only if for some $n \in \N_{>0}$ there exist $s_0, s_1, \dots, s_n \in S$ such that $s_0 = x$, $s_n = y$, and:

\(\ds s_0\) | \(\RR\) | \(\ds s_1\) | ||||||||||||

\(\ds s_1\) | \(\RR\) | \(\ds s_2\) | ||||||||||||

\(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||

\(\ds s_{n - 1}\) | \(\RR\) | \(\ds s_n\) |

### Union of Compositions

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

Let:

- $\RR^n := \begin {cases} \RR & : n = 1 \\ \RR^{n-1} \circ \RR & : n > 1 \end {cases}$

where $\circ$ denotes composition of relations.

Finally, let:

- $\ds \RR^+ = \bigcup_{i \mathop = 1}^\infty \RR^i$

Then $\RR^+$ is called the **transitive closure** of $\RR$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Transitive Closures"

The following 11 pages are in this category, out of 11 total.

### E

### T

- Transitive Closure Always Exists
- Transitive Closure is Closure Operator
- Transitive Closure of Reflexive Relation is Reflexive
- Transitive Closure of Reflexive Symmetric Relation is Equivalence
- Transitive Closure of Relation Always Exists
- Transitive Closure of Set-Like Relation is Set-Like
- Transitive Closure of Symmetric Relation is Symmetric