  Equilibrium properties of long-range interacting systems on lattices are
investigated. There was a conjecture by Cannas et. al. that the mean-field
theory is exact for spin systems with non-additive long-range interactions.
This is called "exactness of the mean-field theory". We show that the exactness
of the mean-field theory holds for systems on a lattice with non-additive two
body long-range interactions in the canonical ensemble with non-fixed order
parameters. We also show that in canonical ensemble with fixed order parameters
(e.g. lattice gas model with a fixed number of particles), exactness of the
mean-field theory does not hold in some parameter region, which we call "non-MF
region". In the non-MF region, an inhomogeneous configuration appears contrary
to the uniform configuration in the region where the mean-field theory holds.
This inhomogeneous configuration is not the one given by the standard phase
separation. Therefore, the mean-field picture is not adequate to describe these
states. We discuss phase transitions between the MF region and the non-MF
region. Exactness of the mean-field theory in spin glasses is also discussed.
