On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the "triplet order parameter" 〈σ_iσ_jσ_k〉 are the same as those of the magnetization 〈σ_i〉. Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperature T_c, defined as the temperature below which there is a spontaneous magnetization, is for fixed "even" spin interactions J_e independent of the way in which the "odd" interactions J_o approach zero from "above." This is achieved by using only the "simplest," Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.