Consider the relation:
$$\langle x_1,x_2\rangle\prec\langle y_1,y_2\rangle\iff x_1,x_2,y_1,y_2\in\Bbb N\wedge x_1y_2<x_2y_1.$$
This is usually used for defining the (positive) fractions $\Bbb Q^+$ as part of the construction of the reals in set theory. What interests me, though, is that this relation has a lot more properties than you would expect just from being a partial order:
- It is a (strict) partial order
- Defining $x\approx y\iff x\not\prec y\wedge y\not\prec x$, $\approx$ is an equivalence relation
- $\prec$ is compatible with $\approx$-equivalence: if $x\approx y$ then $z\prec x$ iff $z\prec y$ and $x\prec z$ iff $y\prec z$.
- Defining $<$ as a function on equivalence classes as $[x]<[y]\iff x\prec y$, $<$ is a total order
Is there a characterization of this kind of relation? I'm pretty sure that most of the properties above follow from just a small set of extra requirements, but this list definitely has non-partial order properties in it, for example using $\subset$ as the partial order: $\{1\}\approx\{2\}\approx\{1,3\}\supset\{1\}$ violates the second bullet. Does the last bullet follow from the first three?
Edit: After thinking a bit about how to prove the second bullet, I came up with the necessary condition $$x\prec y\to x\prec z\vee z\prec y.\tag{$\ast$}$$ Is there a name for equation $(\ast)$ or partial orders that satisfy $(\ast)$?