The Hammersley-Clifford theorem and lattice supported distributions

For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also discuss the interesting history of the original theorem and show a connection to the Hibi ideals of lattices.