  The decomposition into interaction subspaces is a hierarchical decomposition
of the spaces of cylindrical functions of a finite product space, also called
factor spaces. It is an important construction in graphical models and a
standard way to prove the Hammersley-Clifford theorem that relates Markov
fields to Gibbs fields and plays a central role in Kellerer's result for the
linearized marginal problem. We define an intersection of sum property, or
simply intersection property, and show that it characterizes collections of
vector subspaces over a poset that can be hierarchically decomposed into direct
sums, giving therefore a general setting for such construction to hold. We will
call this generalization the interaction decomposition. The intersection
property is the Bayesian intersection property when specified to factor spaces
which, under this new perspective on the interaction decomposition, appears to
be a structure property. An application is the extension of the decomposition
into interaction subspaces for any product of any set.
