  The ground state of interacting particles on a disordered one-dimensional
host-lattice is studied by a direct numerical method. It is shown that if the
concentration of particles is small, then even a weak disorder of the
host-lattice breaks the long-range order of Generalized Wigner Crystal,
replacing it by the sequence of blocks (domains) of particles with random
lengths. The mean domains length as a function of the host-lattice disorder
parameter is also found. It is shown that the domain structure can be detected
by a weak random field, whose form is similar to that of the ground state but
has fluctuating domain walls positions. This is because the generalized
magnetization corresponding to the field has a sufficiently sharp peak as a
function of the amplitude of fluctuations for small amplitudes.
