  Within the recently proposed doped-carrier representation of the projected
lattice electron operators we derive a full Ising version of the t-J model.
This model possesses the global discrete Z_2 symmetry as a maximal spin
symmetry of the Hamiltonian at any values of the coupling constants, t and J.
In contrast, in the spin anisotropic limit of the t-J model, usually referred
to as the t-J_z model, the global SU(2) invariance is fully restored at J_z=0,
so that only the spin-spin interaction has in that model the true Ising form.
We discuss a relationship between those two models and the standard isotropic
t-J model. We show that the low-energy quasiparticles in all three models share
the qualitatively similar properties at low doping and small values of J/t. The
main advantage of the proposed Ising t-J model over the t-J_z one is that the
former allows for the unbiased Monte Carlo calculations on large clusters of up
to 10^3 sites. Within this model we discuss in detail the destruction of the
antiferromagnetic order by doping as well as the interplay between the AF order
and hole mobility. We also discuss the effect of the exchange interaction and
that of the next nearest neighbour hoppings on the destruction of the AF order
at finite doping. We show that the short-range AF order is observed in a wide
range of temperatures and dopings, much beyond the boundaries of the AF phase.
We explicitly demonstrate that the local no double occupancy constraint plays
the dominant role in destroying the magnetic order at finite doping. Finally, a
role of inhomogeneities is discussed.
