# Definition:Transitive Closure (Relation Theory)

## Definition

### Definition 1

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.

### Definition 2

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

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

### Definition 3

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

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

For $x, y \in S$, $x \mathrel {\mathcal R^+} y$ 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:

\(\ds s_0\) | \(\mathcal R\) | \(\ds s_1\) | ||||||||||||

\(\ds s_1\) | \(\mathcal R\) | \(\ds s_2\) | ||||||||||||

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

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

### Definition 4

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

Let:

- $\mathcal R^n := \begin{cases} \mathcal R & : n = 0 \\ \mathcal R^{n-1} \circ \mathcal R & : n > 0 \end{cases}$

where $\circ$ denotes composition of relations.

Finally, let:

- $\displaystyle \mathcal R^+ = \bigcup_{i \mathop \in \N} \mathcal R^i$

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

## Also known as

Some authors refer to the **transitive closure** of $\mathcal R$ as the **ancestral** of $\mathcal R$.

The symbolism varies: some authors use $\mathcal R^t$. The literature is inconsistent, so any notation needs explanation when used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Results about
**reflexive closures**can be found here. - Equivalence of Definitions of Transitive Closure (Relation Theory)
- Recursive Construction of Transitive Closure

## Sources

- 1967: Saunders Mac Lane and Garrett Birkhoff:
*Algebra*: $\S 1.3$: Exercise $4$