Structure of ground states in three-dimensional using model with three-step interaction

The recent work [1] by S.  A. Pirogov and Ya.  G. Sinay investigated the phase diagrams for classical lattice systems with finite number of ground states, which satisfy a certain stability condition. This condition was called the Payerls condition in the work [1]. For corresponding Hamiltonians it was proved that the structure of the phase diagrams is determined by the structure of ground states. Thus the problem of studying the phase diagrams was reduced to the problem of investigating the ground states of the original Hamiltonians. Structure of ground states for three-dimensional Ising model with the two-step interaction is given in the work [2] by V.  M. Gertsik and R.  L. Dobrushin. The present work investigates the structure of ground states and tests the Payerls condition for certain Hamiltonians of the Ising type. Some generalizations are presented in the last section of the paper.