  Using an extended Lee-Yang theorem and GKS correlation inequalities, we
prove, for a class of ferromagnetic multi-spin interactions, that they will
have a phase transition(and spontaneous magnetization) if, and only if, the
external field $h=0$ (and the temperature is low enough). We also show the
absence of phase transitions for some nonferromagnetic interactions. The FKG
inequalities are shown to hold for a larger class of multi-spin interactions.
