are interested in, is actually a sum of the products of many competitive ferromagnetic, antiferromagnetic and more exotic interactions. The thermodynamic properties of the system, described by the Hamiltoniam $\overline{H}$ (24), are then a result of the competition between all of them. Which interaction wins in such a competition it certainly depends on temperature, model parameters ($%
t,U$) and on the average number of electrons per lattice site, determining the chemical potential of the system.

The formalism presented in this paper is also applicable to a more complicated decomposition of the Hubbard Hamiltonian (1) into a set od interacting clusters consisting e.g. of one central atom and $z$ its nearest neighbours. We, however, know (see e.g. Refs \[36\]-\[39\] and papers cited therein) that, unfortunately, the mathematical problems in this case exponentially grows up with the size of the cluster.

Taylor expansion
================

The complicated form of the Hubbard Hamiltonian $\overline{H}$ in the large $%
U$ limit (see (24), (C.6)) where $P$ is given by (22) (see also (17)) can essentially be reduced when applying the Taylor expansion with respect to the parameter $x=\frac{t}{U}\ll 1$. To do it we have to expand all the coefficients in (14), (17) and (C.6) including 