Local finiteness in varieties of MS4-algebras

2023/12/28 by Bezhanishvili, Guram, Meadors, Chase · 1 citation
#03B45 #06E15 #06E25 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2312.16754

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 MS4_S has the finite model property. We show that both S5² and S4_u are proper extensions of MS4_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 S5₂-algebras into varieties of 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 MS4-algebras (even of MS4_S-algebras) is at least as hard as that of characterizing locally finite varieties of S5₂-algebras -- a problem that remains wide open.

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