Negative equivalence over the minimal logic and interpolation

It is proved that extensions of the minimal Johansson logic J are negatively equivalent if and only if their centers are equal. It is proved in [1] that the logics with the weak interpolation property WIP are divided into eight intervals with etalon logics on the top. Therefore a logic possesses WIP iff it is negatively equivalent to one of the eight etalon logics. An axiomatization and a semantic characterization are found for WIP-minimal logics, which are the least elements of all eight intervals of logics with WIP. The Craig interpolation property CIP is stated for the most of WIP-minimal logics.