Interpolation over the minimal logic and Odintsov intervals

We study Craig's interpolation property in the extensions of Johansson's minimal logic. We consider the Odintsov classification of J-logics according to their intuitionistic and negative companions which subdivides all logics into intervals. We prove that the lower endpoint of an interval has Craig interpolation property if and only if both its companions do so. We also establish the recognizability of the lower and upper endpoints which have Craig interpolation property, and find their semantic characterization.