On ferromagnetism and antiferromagnetism

Considering spin lattice systems interacting through arbitrarily many-body and indefinitely-long-range interactions, we look at geometric or algebraic conditions to ensure a convergent time development. We pone positivity or semi-boundedness requirements for induced Hamiltonians in the extremal « semi-symmetric » states, using the adjacency of two intertwining lattices, as Neel did, to describe ferromagnetic and antiferromagnetic behaviours. We essentially deal with spin 1/2, but rough indications are given how to manage the results when arbitrary spins are involved.