Some characterization and preservation theorems in modal logic

A class of Kripke models is modally definable if there is a set of modal formulas such that the class consists exactly of models on which every formula from that set is globally true. In this paper the notion of modal definability is generalized in the following way: a class is also considered definable if there is a set of formulas such that it consists exactly of models in which every formula from that set is satisfiable. By generalizing this approach, various types of modal definability on the level of Kripke models are considered and characterization theorems in the usual form in terms of algebraic closure conditions are given. As some consequences of these, various preservation results are presented. Also, some characterizations are strenghtened by replacing closure under ultraproducts with closure under ultrapowers.