Subdirectly Irreducible Modal Algebras and Initial Frames

This paper is centred around a nice conjecture, known to Wolfgang Rautenberg and myself since 1978: a modal algebra A is subdirectly irreducible if and only if its dual frame A∗ is initial, or generated (see definitions in the text). I here show that in full generality the conjecture is false, but that it becomes true under some mild additional assumptions. Unfortunately, some counterexamples seem to suggest that there is no room for just one general result containing all cases in which the conjecture can be proved. Still, some of the theorems here proved are general enough to include such interesting cases as the finite case, the case of Kripke frames and the case of modal logic K4. The literature on the subject consists, to my knowledge, exclusively of pages 154-156 of Rautenberg’s book [Rautenberg 1979]. They contain what I call Rautenberg’s criterion for a modal algebra to be subdirectly irreducible (see 1.11 below) and a reformulation of the conjecture in the special case of logics K , which however seems to be false (see counterexample in Fig. 2 below). Thus the results given here are all new. They can be seen as an application of the ground work made in [Sambin-Vaccaro 1988]; in fact, topology seems to be necessary to be able to express the additional conditions under which the conjecture becomes true. A version of the present paper was sent in 1985 as a letter to Johan van Benthem answering to his curiosity about the conjecture, and circulated privately since then. Due to my change of interests, I lost trace of it. Destiny has given me a second chance only eleven years later, by means of Marcus Kracht. He came across a copy of my manuscript in Berlin, he realised that some of its results covered a gap in his book [Kracht 1998] and thus got in contact with me; in this way the paper was rescued. With remarkable patience, he also saved it from a second oblivion: he convinced me to publish it and assisted me in many ways to revise it (in particular, he rectified the statements of 2.1 and 3.8 and