  Unlike in classical modal logic, in non-classical modal logics the box and
diamond operators frequently fail to be interdefinable. Instead, these logics
impose some compatibility conditions which tie the box and diamond operators
together and ensure that in terms of Kripke semantics they arise from the same
accessibility relation. This is, for instance, the case in the intuitionistic
modal logic of Fischer Servi and the positive modal logic of Dunn. In these
logics, however, such compatibility conditions also impose further conditions
on the accessibility relation. In this paper, we identify the basic
compatibility conditions which ensure that modal operators in distributive
modal logics arise from a single accessibility relation without imposing any
restrictions on the relation. As in the distributive logic of Gehrke,
Nagahashi, and Venema, we allow for negative box and diamond operators here in
addition to the usual positive ones. Intuitionistic modal logic and positive
modal logic, or more precisely the corresponding classes of algebras, are then
obtained in a modular way by adding certain canonical axioms which we call
locality conditions on top of these basic compatibility conditions.
