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.