Is the minimum of two supermodular functions supermodular? Suppose $f$ and $g$ are supermodular, non-negative and increasing set functions. Is the function $$h(X)=\min(f(X),g(X))$$ supermodular? What about the maximum?
 A: This is a counterexample to the $\textit{min}$ question.
Consider the universe $X=\{1,..,5\}.$ 
Let $f,g \in \mathbb{Z}^{\mathcal{P}(X)}$ be defined as follows: $$ f(S)= (\lvert X \rvert + \lvert S \rvert) \cdot \mathbb{1}_{\lvert S \rvert \geq 3 } \hspace{2mm} and \hspace{2mm} g(S)=\lvert S \rvert $$. 
With $A=\{1,2,3\}$ and $B=\{1,2,4\}$, the inequality $$ 6  = min(f(A),g(A)) +  min(f(B),g(B)) \leq min(f(A\cap B),g(A \cap B))+ min (f(A \cup B), g(A \cup B)) = 4 $$ is untrue.  
This is a counterexample to the $\textit{max}$ question. Consider the universe $X=\{1,2,3\}.$ Let $f,g \in \mathbb{Z}^{\mathcal{P}(X)}$ be defined as follows: $$f(S)=\mathbb{1}_{\{1 \in S\}} \hspace{2mm} and \hspace{2mm} g(S)= \mathbb{1}_{\{3 \in S\}}.$$ 
Now let $A=\{1,2\}$ and $B=\{2,3\}$. 
The inequality $$ 2 = max(f(A),g(A)) +  max(f(B),g(B)) \leq max(f(A\cap B),g(A \cap B))+ max (f(A \cup B), g(A \cup B)) = 1 $$ is untrue. 
Consequently, the answer to both questions is no. (I assumed that you meant 'non-decreasing' instead of 'increasing'.)
In 'Submodular functions and convexity', L. Lovász gives a sufficient condition for the minimum of two submodular set functions to be submodular: if $f$ and $g$ are submodular set functions such that $f-g$ is either monotone increasing or decreasing, then $min(f,g)$ is also submodular. The condition applies to supermodularity by substituting supermodular for submodular and $\textit{max}$ for $\textit{min}$. 
