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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. I think the axiomatic geometry guys who study (weakened versions of) Hilbert's fourth problem may have some interesting things to say about this. – Willie Wong Feb 1 '13 at 12:12

There is already something called "betweeness relation". What's more, it seems to be exactly what you want.

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The planetmath page on betweenness relations has moved. – Jacob Maibach Feb 16 '15 at 18:33

For my intermediate question about generalizing partial orders to something where order-preserving and order-reversing maps would both be homomorphisms, here is a sort of brute-force way to make a poset forget its orientation:

We consider a four-way relation $ab\sim cd$, intuitively meaning "the relation between $a$ and $b$ is the same as the relation between $c$ and $d$", and formally defined by $$ ab\sim cd ~\equiv~ (a<b\land c<d)\lor(b<a\land d<c) $$ Celarly, if a partial order $P$ has at least one pair of different but comparable elements, then the $\sim$-preserving maps from $P$ to another partial order are exactly the order-preserving maps plus the order-reversing maps. The same is also (somewhat vacuously) true if $P$ is the trivial partial order.

We can give axioms for $\sim$ that guarantee that it derives from a partial order:

  1. Outer symmetry: $ab\sim xy ~\Rightarrow~ xy\sim ab$
  2. Outer transitivity: $ab\sim xy \land xy\sim pq ~\Rightarrow~ ab\sim pq$
  3. Inner symmetry: $ab\sim xy ~\Rightarrow~ ba\sim yx$
  4. Inner transitivity: $ab\sim bc ~\Rightarrow~ ab\sim ac$
  5. Inner irreflexivity: $\neg(ab\sim xx)$
  6. Connectedness: $ab\sim xy \land pq\sim rs ~\Rightarrow~ ab\sim pq \lor ab\sim qp$

For any $\sim$ that satisfies these axioms we can derive a partial order that induces it: If $\sim$ is the empty relation, then the trivial partial order works. Otherwise choose fixed $a$ and $b$ such that $ab\sim pq$ for some $p$ and $q$ and define $x<y$ to mean $ab\sim xy$. Then $x<y$ is a strict partial order, and $\sim$ is the quaternary relation induced by it.

But elegant it ain't. In particular the connectedness axiom looks rather ad-hoc. (On the other hand, this could be a sign that deleting this axiom might lead to something interesting beyond just posets).

I'm still curious whether something interesting can be done with the "betweenness" idea, though.

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Not enough points to comment, so I post this as an answer:

Maybe the thing you want is a partial cyclic order. See the paper It seems that there is not much known about these relations.

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Concepts along these lines have certainly been studied over a long period of time. A good place to look is S. Adeleke and P. M. Neumann Relations related to betweenness: their structure and automorphisms, Memoirs of the AMS, Vol. 131 No. 623 (1998). They consider various structures of this sort, ($B$-sets and betweenness relations), as well as their relationship with semilinearly ordered sets, a natural structure arising from the maximal chains of semilinearly ordered sets, and another structure on the set of ends of a $B$-set.

Another keyword to look up is pretree. Brian Bowditch for one has studied them (they arise in his study of the boundary of hyperbolic groups). See his memoir Treelike structures arising from continua and convergence groups, Memoirs of the AMS, Vol. 139 No. 662 (1999).

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