final form of the obtained Hamiltonian in the large $U$ limit (see (23), (24) and (C.6)) visualizes high degree of complexity of the model, deeply hidden in its original version, forming a mixture of many multiple ferromagnetic, antiferromagnetic and more complex interactions competing one with another. This fact seems to be decisive for our final conclusion. Because we are still dealing with the approximate solutions of the model (the exact solution does not exist till now) it may happen that we underestimate in this way some important interactions and overestimate the others. It is the reason why the resulting thermodynamic properties of the Hubbard model, obtained in an approximate way, so strongly depend on the quality of applied approximations.

We note in passing that the exact dimer solution may serve as a novel alloy analogy for the Hubbard model, which could be treated by coherent potential approximation (CPA). It is well-known that the standard alloy analogy, based on the atomic limit, does not allow for ferromagnetism in the Hubbard model. This may change by application of the dimer solution which already accounts for a restricted hopping of the band electrons. A corresponding study is in preparation.

Another example of a 