Local finiteness in varieties of MS4-algebras

Abstract

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.

Figures (6)

• Figure 3.1: The dual algebra of $\mathfrak{F}$ belongs to $\mathbf{S4}_u$ but not to $\mathbf{S5}^2$, and the opposite holds for the dual algebra of $\mathfrak{G}$.
• Figure 4.1: The lattice of subvarieties of $\mathbf{S4}$, dually isomorphic to the lattice of normal extensions of $\mathsf{S4}$. The locally finite varieties are precisely the ones below the dotted line.
• Figure 5.1: Descriptive frame of \ref{['hmt_compact']}.
• Figure 6.1: Constructing an $\mathsf{MS4_S}[2]$-frame from an $\mathsf{S5}_2$-frame. In the $\mathsf{S5}_2$-frame on the left, $E_1$-clusters are horizontal lines while $E_2$-clusters are blue vertical lines. In the $\mathsf{MS4_S}[2]$-frame on the right, $R$-clusters are horizontal lines, proper $R$-arrows are drawn with arrowheads, and $E$-clusters are given by the blue rectangles.
• Figure 6.2: Lifting a correct partition of $\mathfrak{F}$ to one of $T(\mathfrak{F})$, as in \ref{['lem:partition-lift']}: $E$-classes are depicted in blue as $E_2$-classes of $\mathfrak{F}$ with the $E_2$-class itself above on the top rail. The correct partition $\widehat{K}$ is depicted in red.

Theorems & Definitions (83)

• Definition 2.1: FS77
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9