  In this paper we develop a comprehensive study on principal eigenvalues and
both the (weak and strong) maximum and comparison principles related to an
important class of nonlinear systems involving fractional $m$-Laplacian
operators. Explicit lower bounds for principal eigenvalues of this system in
terms of the diameter of $\Omega$ are also proved. As application, given
$\lambda,\mu\geq 0$ we measure explicitly how small has to be
$\text{diam}(\Omega)$ so that weak and strong maximum principles associated to
this problem hold in $\Omega$.
