  A generic lattice model for systems containing particles interacting with
short-range attraction long-range repulsion (SALR) potential that can be solved
exactly in one dimension is introduced. We assume attraction J_1 between the
first neighbors and repulsion J_2 between the third neighbors. The ground state
of the model shows existence of two homogeneous phases (gas and liquid) for
J_2/J_1<1/3. In addition to the homogeneous phases, the third phase with
periodically distributed clusters appears for J_2/J_1>1/3. Phase diagrams
obtained in the self-consistent mean-field approximation for a range of values
of J_2/J_1 show very rich behavior, including reentrant melting, and
coexistence of two periodic phases (one with strong and the other one with weak
order) terminated at a critical point. We present exact solutions for the
equation of state as well as for the correlation function for characteristic
values of J_2/J_1. Based on the exact results, for J_2/J_1>1/3 we predict
pseudo-phase transitions to the ordered cluster phase indicated by a rapid
change of density for a very narrow range of pressure, and by a very large
correlation length for thermodynamic states where the periodic phase is stable
in mean field. For 1/9<J_2/J_1<1/3 the correlation function decays
monotonically below certain temperature, whereas above this temperature
exponentially damped oscillatory behavior is obtained. Thus, even though
macroscopic phase separation is energetically favored and appears for weak
repulsion at T=0, local spatial inhomogeneities appear for finite T. Monte
Carlo simulations in canonical ensemble show that specific heat has a maximum
for low density \rho that we associate with formation of living clusters, and
if the repulsion is strong, another maximum for \rho = 1/2.

Soft Condensed Matter;Statistical Mechanics