  Mixture models on order relations play a central role in recent
investigations of transitivity in binary choice data. In such a model, the
vectors of choice probabilities are the convex combinations of the
characteristic vectors of all order relations of a chosen type. The five
prominent types of order relations are linear orders, weak orders, semiorders,
interval orders and partial orders. For each of them, the problem of finding a
complete, workable characterization of the vectors of probabilities is
crucial---but it is reputably inaccessible. Under a geometric reformulation,
the problem asks for a linear description of a convex polytope whose vertices
are known. As for any convex polytope, a shortest linear description comprises
one linear inequality per facet. Getting all of the facet-defining inequalities
of any of the five order polytopes seems presently out of reach. Here we search
for the facet-defining inequalities which we call primary because their
coefficients take only the values -1, 0 or 1. We provide a classification of
all primary, facet-defining inequalities of three of the five order polytopes.
Moreover, we elaborate on the intricacy of the primary facet-defining
inequalities of the linear order and the weak order polytopes.
