Submodularity of the product of two non-negative, monotone increasing submodular functions

Question

I'm trying to prove the submodularity of the product of two non-negative, monotone increasing submodular functions

Formally, we have $f$ and $g$ are submodular functions, that is, $f:2^{\Omega}\rightarrow \mathbb{R}$ and for every $S, T \subseteq \Omega$ we have that $f(S)+f(T)\geq f(S\cup T)+f(S\cap T)$.

We also have that $f$ and $g$ are non-negative: $f(.) \geq 0$

And are monotone increasing: $f(S) \leq f(T)$, for all sets $S \subseteq T$

I just wonder is there a way we can prove that $fg$ is submodular (or not)?

Answer 1

Counter example:

$\Omega = \{a,b\}$. $f(\phi)=0$, $f(\{a\})=1$, $f)\{b\})=2$, $f(\{a,b\}) = 3$.

Multiply $f$ with itself. The result is not submodular.