Types of Frames
In full generality, general frames are hardly more than a fancy name for Kripke models; in particular, the correspondence of modal axioms to properties on the accessibility relation is lost. This can be remedied by imposing additional conditions on the set of admissible valuations.
A frame is called
- differentiated, if implies ,
- tight, if implies ,
- compact, if every subset of V with the finite intersection property has a non-empty intersection,
- atomic, if V contains all singletons,
- refined, if it is differentiated and tight,
- descriptive, if it is refined and compact.
Kripke frames are refined and atomic. However, infinite Kripke frames are never compact. Every finite differentiated or atomic frame is a Kripke frame.
Descriptive frames are the most important class of frames because of the duality theory (see below). Refined frames are useful as a common generalization of descriptive and Kripke frames.
Read more about this topic: General Frame
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