Failure of interpolation in relevant logics

Craig's interpolation theorem fails for the prepositional logicsE of entailment,R of relevant implication andT of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.