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Think about the graph whose nodes are total orders on a finite set, and whose edges connect orders that only differ on two elements. This is actually a Cayley graph of $S_n$, but I don't want to fix a preferred origin (let me abuse notation and call it a Cayley graph anyways). One condition that comes up in Coxetery stuff all the time is that one order $<$ is on a geodesic from $<'$ to $<''$; this is equivalent to there being no pair $a,b$ with $a< b, a>'b$ and $a>''b$. Unfortunately, I've never seen a name given to this condition on total orders, even though it's a very natural notion. It would be helpful for me to have one.

Have you seen this notion defined anywhere? Does it have a consensus name?

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I don't think this is a consensus name, but it would seem natural to me to say that $\le$ interpolates $\le'$ and $\le''$. –  joriki Oct 30 '12 at 7:12
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