A generalization of modal frame definability

A class of Kripke frames is called modally definable if there is a set of modal formulas such that the class consists exactly of frames on which every formula from that set is valid, i.e. globally true under any valuation. Here, existential definability of Kripke frame classes is defined analogously, by demanding that each formula from a defining set is satisfiable under any valuation. The notion of modal definability is then generalized by combining these two. Model theoretic characterizations of these types of definability are given.