Definition:Transitive Closure (Relation Theory)

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 $$\mathcal{R} = \mathcal{R}^+$$.

Note
The symbolism varies: some authors use $$\mathcal{R}^t$$. The literature is inconsistent.