Is there a differentiable smooth max and min function that respects the distributive law?

Question

I'm looking for functions $S_{max}, S_{min}$ that is similar (converges to?) to a smooth max, but respects the distributive law and is differentiable (or is polynomial). Is there such a known function? So it should at least satisfy the following:

1. $S_{max}(x, y)'$ exists assuming that $x', y'$ do.
2. $min(max(x, y), z) = max(min(x, z), min(y, z))$ and vice versa (distributive)