Chase Meadors | Searcharxiv

arXiv2024

Local tabularity in MS4 with Casari's axiom

We study local tabularity (local finiteness) in some extensions of $\mathsf{MS4}$ (monadic $\mathsf{S4}$). Our main result is a semantic characterization of local finiteness in varieties of $\mathsf{M^{+}S4}$-algebras, where $\mathsf{M^{+}S4}$ denotes the extension of $\mathsf{MS4}$ by the Casari axiom. We improve this to a syntactic criterion via the reducible path property identified in [Shap16], and note that the product logic $\mathsf{S4}[n] \times \mathsf{S5}$ is an extension of $\mathsf{M^{+}S4}$, obtaining a criterion for extensions of $\mathsf{S4}[n] \times \mathsf{S5}$ as an application. Next, we give a characterization of local finiteness in varieties of $\mathsf{MS4B}[2]$-algebras, where $\mathsf{MS4B}$ denotes the extension of $\mathsf{MS4}$ by the Barcan axiom. We demonstrate that our methods cannot be extended beyond depth 2, as we give a translation of the fusion $\mathsf{S5}_2$ into $\mathsf{MS4B}[3]$ for $n \geq 3$ that preserves and reflects local finiteness, suggesting that a characterization there remains difficult. Finally, we also establish the finite model property for some of these logics which are not known to be locally tabular.

math.LO↗

arXiv2023

Local finiteness in varieties of MS4-algebras

It is a classic result of Segerberg and Maksimova that a variety of $\mathsf{S4}$-algebras is locally finite iff it is of finite depth. Since the logic $\mathsf{MS4}$ (monadic $\mathsf{S4}$) axiomatizes the one-variable fragment of $\mathsf{QS4}$ (predicate $\mathsf{S4}$), it is natural to try to generalize the Segerberg--Maksimova theorem to this setting. We obtain several results in this direction. Our positive results include the identification of the largest semisimple variety of $\mathsf{MS4}$-algebras. We prove that the corresponding logic $\mathsf{MS4_S}$ has the finite model property. We show that both $\mathsf{S5}^2$ and $\mathsf{S4}_u$ are proper extensions of $\mathsf{MS4_S}$, and that a direct generalization of the Segerberg--Maksimova theorem holds for a family of varieties containing the variety of $\mathsf{S4}_u$-algebras. Our negative results include a translation of varieties of $\mathsf{S5}_2$-algebras into varieties of $\mathsf{MS4_S}$-algebras of depth 2, which preserves and reflects local finiteness. This, in particular, shows that the problem of characterizing locally finite varieties of $\mathsf{MS4}$-algebras (even of $\mathsf{MS4_S}$-algebras) is at least as hard as that of characterizing locally finite varieties of $\mathsf{S5}_2$-algebras -- a problem that remains wide open.

math.LO↗