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A homeomorphism $\mathbb R\to\mathbb R$ is almost the same thing as an order isomorphism, except that a homeophorphism can also be an order anti-isomorphism.

I'm wondering whether there is a natural first-order structure "X" which generalizes partial orders (in the sense that order-presering and order-reversing maps would be the prototypical examples of "X morphisms") such that the homeomorphisms $\mathbb R\to\mathbb R$ are exactly the "X isomorphisms".

So far the most promising approach seems to be consider a trinary "betweenness" relation $$\beta(a,b,c) \equiv (a\le b\le c) \lor (c\le b\le a)$$ and look at the category of $\beta$-preserving maps.

Have such structures been studied? Do they have a name? Is there a nice axiomatic characterization of the trinary relations that can be induced by a partial (or total?) order in this way?

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    I think some of these are studied under the heading of [ordered geometry](http://en.wikipedia.org/wiki/Ordered_geometry). I think the axiomatic geometry guys who study (weakened versions of) Hilbert's fourth problem may have some interesting things to say about this.2013-02-01

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