  We propose a new framework of the reflection positivity in terms of order
preserving operator inequalities. From the viewpoint of the new framework, we
study one-dimensional fermion-phonon systems. Especially, properties of ground
states are examined in detail. It is clarified that the reflection positivity
consistently describes the charge-density-wave (CDW) order in this system.
Furthermore, if the Coulomb interaction is long ranged, then we prove the
existence of the CDW long range order in the ground state.
