Local finiteness in varieties of MS4-algebras

Abstract

It is a classic result of Segerberg and Maksimova that a variety of $S4$-algebras is locally finite iff it is of finite depth. Since the logic $MS4$ (monadic $S4$) axiomatizes the one-variable fragment of $QS4$ (predicate $S4$), 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 $MS4$-algebras. We prove that the corresponding logic $\mathsf{MS}{4}_{\mathsf{S}}$ has the finite model property. We show that both ${S5}^2$ and ${S4}_u$ are proper extensions of $\mathsf{MS}{4}_{\mathsf{S}}$, and that a direct generalization of the Segerberg--Maksimova theorem holds for a family of varieties containing the variety of ${S4}_u$-algebras. Our negative results include a translation of varieties of ${S5}_2$-algebras into varieties of $\mathsf{MS}{4}_{\mathsf{S}}$-algebras of depth 2, which preserves and reflects local finiteness. This, in particular, shows that the problem of characterizing locally finite varieties of $MS4$-algebras (even of $\mathsf{MS}{4}_{\mathsf{S}}$-algebras) is at least as hard as that of characterizing locally finite varieties of ${S5}_2$-algebras -- a problem that remains wide open.

arXiv comment28 pages, 7 figures