The diffusion in the multicomponent mixture of molecules coadsorbed on a surface with a square symmetry of the adsorption sites is investigated in the framework of a lattice gas model by a theoretical approach and the Monte Carlo simulations. Using the approach based on the theory of the non-equilibrium statistical operator I have derived the systems of equations describing the diffusion of the molecules in the Fickian and Onsager forms and obtained the analytical expressions for the Fickian diffusivities. These expressions are derived taking into account the lateral interactions between the molecules and the interactions of the activated molecules in the saddle points of the lattice potential relief with their surroundings. The diffusion equations in the Fickian representation exactly coincide with the corresponding classical counterparts, but the Onsager-like transport equations differ from their standard lore. In the new Onsager representation the migration of the diffusants is controlled only by the gradients of their own chemical potentials and the cross terms are absent. The Monte Carlo simulations have been used to test the correctness of the new expressions. There is a very good coincidence between the analytical and numerical data in the whole coverage region and in the wide range of the lateral interactions. The lateral interactions substantially influence on the behavior of the thermodynamic quantities and the Fickian diffusivities. These results demonstrate that the classical description in the Onsager presentation is not universal and cannot be applied for some lattice gas systems. (C) 2019 Elsevier Ltd. All rights reserved.