Normal extensions of KTB of codimension 3

Abstract

It is known that in the lattice of normal extensions of the logic KTB there are unique logics of codimensions 1 and 2, namely, the logic of a single reflexive point, and the logic of the total relation on two points. A natural question arises about the cardinality of the set of normal extensions of KTB of codimension 3. Generalising two finite examples found by a computer search, we construct an uncountable family of (countable) graphs, and prove that certain frames based on these produce a continuum of normal extensions of KTB of codimension 3. We use algebraic methods, which in this case turn out to be better suited to the task than frame-theoretic ones.