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Ground states for nonuniform periodic Ising chains

We give a generalization of Morita's works on ground states of Ising chains, for chains with a periodic structure with different spins, and distant neighbor interactions. The main assumption is translational invariance. The length of the irreducible blocks is a multiple of the period of the chain. In the case of parity invariance, it restricts the length in general only in the diatomic case. There are degenerated states and under certain circumstances there could be nonregular ground states. We illustrate the results and give the ground state diagrams in several cases.