Definition:Transitive Closure (Relation Theory)

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

The transitive closure of $\mathcal R$ is denoted $\mathcal R^+$, and is defined as the smallest transitive relation on $S$ which contains $\mathcal R$.

The transitive closure of $\mathcal R$ always exists.

It is clear that if $\mathcal R$ is itself transitive, then $\mathcal R = \mathcal R^+$.

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 ProofWiki.