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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](http://www.google.com/search?client=safari&rls=en&q=antisymmetric+up+to+equivalence&ie=UTF-8&oe=UTF-8#sclient=psy-ab&hl=en&safe=off&client=safari&rls=en&source=hp&q=%22antisymmetric+up+to+equivalence%22&pbx=1&oq=%22antisymmetric+up+to+equivalence%22&aq=f&aqi=&aql=1&gs_sm=e&gs_upl=14802l15356l2l15567l2l2l0l0l0l0l301l493l0.1.0.1l2l0&bav=on.2,or.r_gc.r_pw.&fp=99477dd25ee5c698&biw=1680&bih=933).2011-09-27

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