As I understand it, a binary relation $R$ over a set $A$ is antisymmetric if for all $a, b \in A: aRb \land bRa$ implies $a = b$.
Now, suppose that I have an equivalence relation $E$ over the set $A$. Is there a term for a relation $R$ over A such that if $aRb$ and $bRa$, then $aEb$? This is similar to the definition of antisymmetry from above, but we use the equivalence relation $E$ instead of straight equality.
Thanks!