  We compute the ground-state fidelity and various correlations to gauge the
competition between different orders in two-dimensional t-J-type models. Using
exact numerical diagonalization techniques, these quantities are examined for
(i) the plain t-J and t-t'-J models, (ii) for the t-J model perturbed by
infinite-range d-wave or extended-s-wave superconductivity inducing terms, and
(iii) the t-J model, plain and with a d-wave perturbation, in the presence of
non-magnetic quenched disorder. Various properties at low hole doping are
contrasted with those at low electron filling. In the clean case, our results
are consistent with previous work that concluded that the plain t-J model
supports d-wave superconductivity. As a consequence of the strong correlations
present in the low hole doping regime, we find that the magnitude of the d-wave
condensate occupation is small even in the presence of large d-wave
superconductivity inducing terms. In the dirty case, we show the robustness of
the ground state in the strongly correlated regime against disorder.
