  Semigraphoids are combinatorial structures that arise in statistical learning
theory. They are equivalent to convex rank tests and to polyhedral fans that
coarsen the reflection arrangement of the symmetric group. We resolve two
problems on semigraphoids posed in Studeny's book, and we answer a related
question by Postnikov, Reiner, and Williams on generalized permutohedra. We
also study the semigroup and the toric ideal associated with semigraphoids.
