Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable.
What does "$f(Ax)$ is supermodular" mean?
For example, if $f(x)$ is supermodular, we have the cross partials of $f()$ with respect to $x_1$ and $x_2$ nonnegative.
How can we similarly represent $f(Ax)$ supermodular in terms of the second partial and cross partial derivatives?
Note: I am interested in the expression for the continuous case and not the general lattice definition if possible.
Thank you for all your help!