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I would like to know if there's a special name for this kind of ordering. When I say there is a strict total order on $X/\sim$, what I mean is that two distinct elements in the same equivalence classes are considered equal, but an element of one equivalence class is either less than or greater than an element of another equivalence class.

In other words, what kind of ordering does the set of poker hands have? For example, one equivalence class is the set of straight flushes with a high card of 10 (i.e., 6 through 10 of the same suit). Any two from these four hands are considered equal. However, a hand from this set is greater than any poker hand that's less than a straight flush (e.g. four of a kind, straight flush, etc.) or a straight flush with a high card less than 10, and less than any straight flush with high card greater than 10.

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$$\setminus\neq/$$ – Asaf Karagila Aug 25 '12 at 10:32
Ok, thanks for noting. – ladaghini Aug 25 '12 at 10:45
up vote 2 down vote accepted

We say that $R$ is a preorder of $X$ if $R$ is reflexive and transitive. Then equivalence relation $x\sim y\iff xRy\land yRx$ has the property that $X/\sim$ is partially ordered.

If the partial order is a total order, then $R$ is a total preorder.

The inverse holds too, if $X/\sim$ is totally ordered by $\leq$ we define $xRy\iff x/\sim\leq y/\sim$. It is a nice exercise to see that this is a total preorder of $X$.

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I was hoping for something more specific than preorder, given that $\sim$ is an equivalence relation on $X$. But I'm guessing $\sim$ has less to do with ordering and more to do with partitioning, so I can't get more specific than a preorder? – ladaghini Aug 28 '12 at 9:14
@ladaghini: Yeah, the equivalence has little to do with the ordering. Preorder is about as good as you get. I suppose you can come up with ad-hoc names which describe a state where every equivalence class has exactly $n$ members... – Asaf Karagila Aug 28 '12 at 9:31

Common terms are complete preorder or total preorder or complete quasi-order or total quasi-order.

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