  The magnetic phase diagram of the quarter-filled generalized Wigner lattice
with nearest- and next-nearest-neighbor hopping t_1 and t_2 is explored. We
find a region at negative t_2 with fully saturated ferromagnetic ground states
that we attribute to kinetic exchange. Such interaction disfavors
antiferromagnetism at t_2 <0 and stems from virtual excitations across the
charge gap of the Wigner lattice, which is much smaller than the Mott-Hubbard
gap proportional to U. Remarkably, we find a strong dependence of the charge
structure factor on magnetism even in the limit U to infinity, in contrast to
the expectation that charge ordering in the Wigner lattice regime should be
well described by spinless fermions. Our results, obtained using the
density-matrix renormalization group and exact diagonalization, can be
transparently explained by means of an effective low-energy Hamiltonian.
