Conditional independence structures examined via minors

The notion of minor from matroid theory is adapted to examination of classes of conditional independence structures. For the classes of semigraphoids, pseudographoids and graphoids, finite sets of their forbidden minors are found. The separation graphoids originating from simple undirected graphs and triangulated graphs are characterized in this way neatly as well. Semigraphoids corresponding to the local Markov property of undirected graphs and to the d-separation in directed acyclic graphs are discussed. A new class of semimatroids, called simple semimatroids, is introduced and an infinite set of its forbidden minors constructed. This class cannot be characterized by a finite number of axioms. As a consequence, the class of all semimatroids and the classes of conditional independence structures of stochastic variables and of linear subspaces have infinite sets of forbidden minors and have no finite axiomatization. The closure operator of semimatroids is examined by linear programming methods. All possibilities of conditional independences among disjoint groups of four random variables are presented.