Some remarks on T-supermodularity of Choquet integral

We prove that total monotonicity of monotone measures is a sufficient (but not necessary) condition for T-supermodularity of Choquet integral. Moreover, we show that total monotonicity does not imply, in general, supermodularity of the integral, when we consider the symmetric Choquet integral or the Sugeno integral. Finally, we also prove that, for the Choquet integral, T-supermodularity implies supermodularity, when T is the product t-norm, and it is equivalent to supermodularity, when T is the Łuckasiewicz t-norm.