Free Modal Algebras Revisited: The Step-by-Step Method

Abstract

We review the step-by-step method of constructing finitely generated free modal algebras. First we discuss the global step-by-step method, which works well for rank one modal logics. Next we refine the global step-by-step method to obtain the local step-by-step method, which is applicable beyond rank one modal logics. In particular, we show that it works well for constructing the finitely generated free algebras for such well-known modal systems as T, K4 and S4. This yields the notions of one-step algebras and of one-step frames, as well as of universal one-step extensions of one-step algebras and of one-step frames. We show that finitely generated free algebras for T, K4 and S4 and their dual spaces can be obtained by iterating the universal one-step extensions of one-step algebras and of one-step frames. In the final part of the paper we compare our construction with recent literature, especially with [11] which has a very similar approach.