Ascending and Descending Conditional Independence Relations

We explore the conditional probabilistic independences of systems of random variables (I; JjK), to read \I is conditionally independent of J given K", which are ascending (I; JjK)) (I; JjK L) or descending (I; JjK L)) (I; JjK). The resulting abstract independence structures can be equivalently described by weak families of connected sets. Using, in addition, probabilistic representations of matroids we present probabilistic representations of the structures and formulate duality conjectures. Local and global forms of semigraphoids and these independence structures are elaborated and special cases including unconditional independence relations, xed{context relations and Markov networks are discussed.