Local Goldblatt-Thomason theorem

The celebrated theorem proved by Goldblatt and Thomason in 1974 gives necessary and sufficient conditions for an elementary (i.e., first-order definable) class of Kripke frames to be modally definable. Here we obtain a local analogue of this result, where in place of frames we consider n-frames, which are frames with n distinguished worlds. For talking about n-frames, we generalize customary modal formulas to what we call modal expressions. Unlike a modal formula, which is evaluated at a single world of a Kripke model, a modal expression with n individual variables is evaluated at an n-tuple of worlds, just as a first-order formula with n free variables. We introduce operations on n-frames that preserve validity of modal expressions, and show that closure under these operations is a necessary and sufficient condition for an elementary class of n-frames to be modally definable. We also discuss the relationship between modal expressions and hybrid logic and leave open questions.