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 $\smash {\mathord {\sim }^{\mathrm{trans}}_{R}}\subset \mathord {\sim }_{S}$.
