The paper by C. M. Fortuin and P. W. Kasteleyn, with contributions from J. Ginibre, explores the correlation inequalities on partially ordered sets, specifically finite distributive lattices. The authors build upon earlier work by Griffiths and Harris, who derived inequalities for the correlations of Ising ferromagnets and percolation models, respectively. The main result generalizes the positivity of correlations to the case where the set is a finite distributive lattice and the measure satisfies a suitable convexity condition. This generalization is shown to be applicable to various physical systems, including Ising spin systems and the random cluster model. The paper also discusses the lattice-theoretic notions and provides applications to these models, demonstrating that the sufficient condition for the positivity of correlations is not necessary.The paper by C. M. Fortuin and P. W. Kasteleyn, with contributions from J. Ginibre, explores the correlation inequalities on partially ordered sets, specifically finite distributive lattices. The authors build upon earlier work by Griffiths and Harris, who derived inequalities for the correlations of Ising ferromagnets and percolation models, respectively. The main result generalizes the positivity of correlations to the case where the set is a finite distributive lattice and the measure satisfies a suitable convexity condition. This generalization is shown to be applicable to various physical systems, including Ising spin systems and the random cluster model. The paper also discusses the lattice-theoretic notions and provides applications to these models, demonstrating that the sufficient condition for the positivity of correlations is not necessary.