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In the work, we investigated a generalized model of the fermionic lattice gasin the form of the extended Hubbard model with intersite Ising-likeinteractions (both antiferromagnetic and ferromagnetic) at the atomic limit onthe triangular lattice. In the ground state, we find the exact phase diagram asa function of $\mu$. Within the mean-field decoupling of the intersite term andexact treatment of onsite interaction, we found also the diagrams for $T\geq 0$including metastable phases. For antiferromagnetic coupling, we find thatnontrival ordered phase can exist with coexistence of charge and metamagneticordering. The transition between the ordered phase and the nonordered phase canbe discontinuous as well continuous depending on the model parameters.Moreover, the ordered phase can coexist with the nonordered phase in phaseseparated states for fixed electron concentration. Additionally, the ranges ofphases metastability are determined in the neighborhood of the discontinuoustransitions.

Magnetic and charge orders on the triangular lattice: Extended Hubbard model with intersite Ising-like magnetic interactions in the atomic limit