 As was the case for PAL, one often wishes to restrict attention to Kripke models whose relations \(R_a\) satisfy certain desirable properties such as reflexivity, transitivity, Euclideanness, and seriality. In order to study actions over such classes, we must be certain that the actions do not transform a Kripke model in the class into a new Kripke model not in the class; that is, we must ensure that the class of Kripke models is “closed” under actions. The following theorem provides some sufficient conditions that guarantee closure. 