Bayesian/Graphoid intersection property for factorisation models

We remark that the Graphoid intersection property, also called intersection property in Bayesian networks, is a particular case of an intersection property, in the sense of intersection of coverings, for factorisation spaces, also called factorisation models, factor graphs. Direct consequences of this are the equivalence between pairwise Markov property and local Markov property, the Hammersley-Clifford theorem.