Degree of Kripke-incompleteness of Tense Logics

The degree of Kripke-incompleteness of a logic $L$ in some lattice $\mathcal{L}$ of logics is the cardinality of logics in $\mathcal{L}$ which share the same class of Kripke-frames with $L$. A celebrated result on Kripke-incompleteness is Blok’s dichotomy theorem for the degree of Kripke-incompleteness in $\mathsf{N}\mathsf{E}\mathsf{x}\mathsf{t}(\mathsf{K})$: every modal logic $L\in \mathsf{N}\mathsf{E}\mathsf{x}\mathsf{t}(\mathsf{K})$ is of the degree of Kripke-incompleteness $1$ or $2^{{\mathrm{\aleph }}_0}$. In this work, we show that the dichotomy theorem for $\mathsf{N}\mathsf{E}\mathsf{x}\mathsf{t}(\mathsf{K})$ can be generalized to the lattices $\text{K}$, $\text{LT}$ and $\text{NExt}(\text{ST})$ of tense logics. We also prove that in $\text{K}$, $\text{LT}$ and $\text{NExt}(\text{ST})$, iterated splittings are exactly the strictly Kripke-complete logics.