An officemate passes along the following natural-seeming question:

Say that a binary operation $\oplus$ is compatible with the usual order $\leq$ on $\mathbb{R}$ if for any $w, x, y, z$ in $\mathbb{R}$ we have $w \leq x$ and $y \leq z$ implies $w \oplus y \leq x \oplus z$. For example, $+$, $\max$ and $\min$ are compatible with $\leq$, as is $\times$ if we restrict to the positive reals. More generally, any operation of the form $a \oplus b = f(g(a) + h(b))$ is compatible with $\leq$ whenever $f$, $g$ and $h$ are increasing functions. (Well, "more generally" is a little bit of a lie, since max and min are actually limits of such expressions.)

Are there any other examples of such binary operations?

(Application of appropriate tags would be appreciated.)