On displaying negative modalities

We extend Takuro Onishi's result on displaying substructural negations by formulating display calculi for non-normal versions of impos- sibility and unnecessity operators, called regular and co-regular negations, respectively, by Dimiter Vakarelov. We make a number of connections be- tween Onishi's work and Vakarelov's study of negation. We also prove a decidability result for our display calculus, which can be naturally extended to obtain decidability results for a large number of display calculi for logics with negative modal operators.