We introduce a new abstract approach to the study of conditional independence, founded on a concept analogous to the factoriza-tion properties of probabilistic independence, rather than the separation properties of a graph. The basic ingredient is the \con-ditional product", which provides a way of combining the objects under consideration while preserving as much independence as possible. We i...

Given a finite simple undirected graph G there is simplicial complex Ind(G), called the independence complex, whose faces correspond to independent sets of G. This well-studied concept because it provides fertile ground for interactions between commutative algebra, theory and algebraic topology. In this paper, we consider generalization complex. [Formula: see text], subset vertex set r-independ...