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The adsorbed atoms exhibit tendency to occupy a triangular lattice formed byperiodic potential of the underlying crystal surface. Such a lattice is formedby, e.g., a single layer of graphane or the graphite surfaces as well as (111)surface of face-cubic center crystals. In the present work, an extension of thelattice gas model to $S=1/2$ fermionic particles on the two-dimensionaltriangular (hexagonal) lattice is analyzed. In such a model, each lattice sitecan be occupied not by only one particle, but by two particles, which interactwith each other by onsite $U$ and intersite $W_{1}$ and $W_{2}$ (nearest andnext-nearest-neighbor, respectively) density-density interaction. Theinvestigated hamiltonian has a form of the extended Hubbard model in the atomiclimit (i.e., the zero-bandwidth limit). In the analysis of the phase diagramsand thermodynamic properties of this model with repulsive $W_{1}>0$, thevariational approach is used, which treats the onsite interaction term exactlyand the intersite interactions within the mean-field approximation. The groundstate ($T=0$) diagram for $W_{2}\leq0$ as well as finite temperature ($T>0$)phase diagrams for $W_{2}=0$ are presented. Two different types of charge orderwithin $\sqrt{3} \times \sqrt{3}$ unit cell can occur. At $T=0$, for $W_{2}=0$phase separated states are degenerated with homogeneous phases (but $T>0$removes this degeneration), whereas attractive $W_2<0$ stabilizes phaseseparation at incommensurate fillings. For $U/W_{1}<0$ and $U/W_{1}>1/2$ onlythe phase with two different concentrations occurs (together with two differentphase separated states occurring), whereas for small repulsive $0<U/W_{1}<1/2$the other ordered phase also appears (with tree different concentrations insublattices). The qualitative differences with the model considered onhypercubic lattices are also discussed.

Charge-Order on the Triangular Lattice: A Mean-Field Study for the Lattice $S=1/2$ Fermionic Gas