All splitting logics in the lattice NEXT(KTB.3'A)

Interpolation in Normal Extensions of the Brouwer Logic

“…Obviously, (SAPK) also holds. All the possible reductions for circular frames are described in [12]. Each circular frame C 2n−1 , n ≥ 2 is reducible to some chain frame Ch n .…”

“…Each circular frame C 2n−1 , n ≥ 2 is reducible to some chain frame Ch n . The p-morphism may be described as gluing 'in half' the circle, see [12], Lemma 15. Further, each chain frame Ch 2n−1 is reducible to the chain frame Ch n , again by gluing 'in half'.…”

“…Further, each chain frame Ch 2n−1 is reducible to the chain frame Ch n , again by gluing 'in half'. A chain frame with an even number of points Ch 2n is reducible to both frames: Ch n and Ch n+1 , see [12], Lemmas 13-14. We may conclude, by superposition of p-morphisms, that eventually each circle frame is reducible to • − −•.…”

Normal extensions of KTB of codimension 3

“…Yet KTB is much less investigated that its transitive cousins, and in fact certain tools working very well for transitive logics (for example, canonical formulas) have no KTB counterparts working nearly as well. Among the articles dealing specifically with KTB and its extensions, Kripke incompleteness in various guises was investigated in [15] and [6], interpolation in [7] and [9], normal forms in [16], and splittings in [17], [10] and [8]. In the present article we focus on the upper part of the lattice of normal (axiomatic) extensions of KTB, or viewed dually, the lower part of the lattice of subvarieties of the corresponding variety of modal algebras.…”