  We investigate the ground state of a one-dimensional lattice system that
hosts two different kinds of excitations (species) which interact with a
power-law potential. Interactions are only present between excitations of the
same kind and the interaction strength can be species-dependent. For the case
in which only one excitation is permitted per site we derive a prescription for
determining the ground state configuration as a function of the filling
fractions of the two species. We show that depending on the filling fractions
compatible or incompatible phases emerge. Furthermore, we discuss in detail the
case in which one species is strongly and the other one weakly interacting. In
this case the configuration of the strongly interacting (strong) species can be
considered frozen and forms an effective inhomogeneous lattice for the other
(weak) species. In this limit we work out in detail the microscopic ground
state configuration and show that by varying the density of the weak species a
series of compatible--incompatible transitions occurs. Finally we determine the
stability regions of the weak species in the compatible phase and compare it
with numerical simulations.
