A generalization of modal definability

Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models is usually called modally definable if there is a set of modal formulas such that a class consists exactly of models on which every formula of that set is globally true, i. e. universally quantified standard translations of these formulas to the corresponding first order language are true. Here, the notion of definability is extended to existentially quantified translations of modal formulas – a class is called modally ∃-definable if there is a set of modal formulas such that a class consists exactly of models on which every formula of that set is satisfiable. A characterization result is given in usual form, in terms of closure conditions on such classes of models.