The equivalence closure1 of $\mathord {\sim }_{R}$ is the relation $\smash {\mathord {\sim }^{\mathrm{eq}}_{R}}$2 satisfying the following universal property:3
- Given another equivalence relation $\mathord {\sim }_{S}$ on $A$ such that $R\subset S$, there exists an inclusion $\smash {\mathord {\sim }^{\mathrm{eq}}_{R}}\subset \mathord {\sim }_{S}$.
1Further Terminology: Also called the equivalence relation associated to $\mathord {\sim }_{R}$.
2Further Notation: Also written $R^{\text{eq}}$.
3Slogan: The equivalence closure of $R$ is the smallest equivalence relation containing $R$.
