Suppose you have a partial order $<_o$ on $R^n$, and you restrict it to a relation $<_r$, where $x <_r y \iff x <_o y \wedge ||x - y|| < r$. Now it might happen that $<_o$ happens to be the transitive closure of $<_r$. In this case I define $<_o$ to be "$r$-smooth." If $<_o$ is $r$-smooth for all $r > 0$, then I say it is "smooth." Which partial orders are smooth?
Also, is there a way to characterize a smooth partial order locally? Something analogous to a gradient of a smooth function.
Example: if we define $<_c$ by $x <_c y$ iff every coordinate of $x$ is less than the corresponding coordinate of $y$, then $<_c$ is smooth.