Pretabular Tense Logics over S4t

A logic $L$ is called tabular if it is the logic of some finite frame and $L$ is pretabular if it is not tabular while all of its proper consistent extensions are tabular. In this work, we study pretabular tense logics in the lattice $\mathsf{NExt}(\mathsf{S4}_t)$ of all extensions of $\mathsf{S4}_t$, tense $\mathsf{S4}$. For all $0<n,m,k,l\leq\aleph_0$, we define the tense logic $\mathsf{S4BP}_{n,m}^{k,l}$ with respectively bounded width, depth and z-degree. We give a full characterization of the set $\mathsf{PTAB}(\mathsf{S4.3}_t)$ of all pretabular logics extending $\mathsf{S4.3}_t$, which entails that there are exactly 5 pretabular logics in $\mathsf{NExt}(\mathsf{S4.3}_t)$. Moreover, by providing a full characterization of $\mathsf{PTAB}(\mathsf{S4BP}^{2,\omega}_{2,2})$ and proving that $|\mathsf{PTAB}(\mathsf{S4BP}^{2,\omega}_{2,3})|=2^{\aleph_0}$, we show the anti-dichotomy theorem for cardinality of pretabular extensions in $\mathsf{NExt}(\mathsf{S4}_t)$: for all cardinal $\kappa$ such that $\kappa\leq{\aleph_0}$ or $\kappa=2^{\aleph_0}$, $|\mathsf{PTAB}(L)|=\kappa$ for some $L\in\mathsf{NExt}(\mathsf{S4}_t)$. It follows that $|\mathsf{PTAB}(\mathsf{S4}_t)|=2^{\aleph_0}$, which answers an open problem concerning the cardinality of $\mathsf{PTAB}(\mathsf{S4}_t)$ raised by Rautenberg in 1979.