Strengthenings of submodularity
There is a complete characterization of coverage functions in terms of such equations. For |X|>3 there are more equations than the ones pointed. Each of these equations can be thought as a constraint on discrete $k^{th}$ derivative.

Monotone increase function if and only if first order discrete derivative is +ve. i.e. $f(B)-f(A)\ge 0$  when $A\subseteq B$. 

Submodularity if and only if second order discrete derivative is -ve. i.e. $(f(A\cup B)-f(B))-(f(A)-f(A\cap B))\le 0$.

Similarly if you have conditions on the next $n$ derivatives you get coverage functions. (I think the signs need to be +ve for even order derivative and -ve for odd order derivative)

Something similar was already known in probability. A coverage function can also be thought as a probability measure (upto a scaling constant). The only reference I was able to dig up was page 439 from Feller's book on probability.