The transitive closure of $\mathord {\sim }_{R}$ is the relation $\smash {\mathord {\sim }^{\mathrm{trans}}_{R}}$1 satisfying the following universal property:2
- Given another transitive relation $\mathord {\sim }_{S}$ on $A$ such that $R\subset S$, there exists an inclusion $\smash {\mathord {\sim }^{\mathrm{trans}}_{R}}\subset \mathord {\sim }_{S}$.
1Further Notation: Also written $R^{\mathrm{trans}}$.
2Slogan: The transitive closure of $R$ is the smallest transitive relation containing $R$.
