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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!

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The expression "antisymmetric up to equivalence" is used in at least some publications. –  joriki Sep 27 '11 at 17:48

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