Abstract
This paper is the concluding part of [1] and [2], and it investigates the inner structure of the lattice Λ(MHA) of all varieties of monadic Heyting algebras. For every n ≤ ω, we introduce and investigate varieties of depth n and cluster n, and present two partitions of Λ(MHA), into varieties of depth n, and into varieties of cluster n. We pay a special attention to the lower part of Λ(MHA) and investigate finite and critical varieties of monadic Heyting algebras in detail. In particular, we prove that there exist exactly thirteen critical varieties in Λ(MHA) and that it is decidable whether a given variety of monadic Heyting algebras is finite or not. The representation of Λ(MHA) is also given. All these provide us with a satisfactory insight into Λ(MHA). Since Λ(MHA) is dual to the lattice NExtMIPC of all normal extensions of the intuitionistic modal logic MIPC, we also obtain a clearer picture of the lattice structure of intuitionistic modal logics over MIPC.
Similar content being viewed by others
References
Bezhanishvili, G., ‘Varieties of monadic Heyting algebras. Part I’ Studia Logica 61 (1998), 367-402.
Bezhanishvili, G., ‘Varieties of monadic Heyting algebras. Part II: Duality theory’ Studia Logica 62 (1999), 21-48.
Bezhanishvili, G., ‘Splitting monadic Heyting algebras’ Report # IS-RR-97-0044F, JAIST, 1977.
Birkhoff, G., Lattice Theory, Providence, R.I., 1967.
Blok, W. J., ‘The lattice of modal logics: an algebraic investigation’ Journal of Symbolic Logic 45 (1980), 221-236.
Bull, R. A., ‘MIPC as the formalization of an intuitionist concept of modality’ Journal of Symbolic Logic 31 (1966), 609-616.
Casari, E., Intermediate Logics, Atti degli incontri di Logica Matematica 1981/1982, Scuola di Specializzazione in Logica Matematica, Università di Siena.
Davey, B. A., ‘On the lattice of subvarieties’ Houston Journal of Mathematics 5 (1979), 183-192.
Esakia, L., and V. Meskhi, ‘Five critical modal systems’ Theoria 43 (1977), Part I, 52-60.
Esakia, L., Heyting Algebras I. Duality Theory (in Russian), Metsniereba Press, Tbilisi, 1985.
GrÄtzer, G., General Lattice Theory, Akademie-Verlag, Berlin, 1978.
Hosoi, T., ‘On intermediate logics I’ Journal of Faculty of Science, University of Tokyo, Sec. I 14 (1967), 293-312.
Hosoi, T., and I. Masuda, ‘A study of intermediate propositional logics on the third slice’ Studia Logica 52 (1993), 15-21.
Hosoi, T., and H. Ono, ‘The intermediate logics on the second slice’ Journal of Faculty of Science, University of Tokyo, Sec. I 17 (1970), 457-461.
Hosoi, T., and H. Ono, ‘Intermediate propositional logics (a survay)’ Journal of Tsuda College 5 (1973), 67-82.
Maksimova, L., ‘Pretabular superintuitionistic logics’ (in Russian), Algebra &; Logic 11 (1972), 558-570.
Maltsev, A., Algebraic Systems (in Russian), Nauka Press, Moscow, 1970.
McKenzie, R., ‘Equational bases and nonmodular lattice varieties’ Transactions of the American Mathematical Society 174 (1972), 1-43.
Monk, D., ‘On equational classes of algebraic versions of logic I’ Mathematica Scandinavica 27 (1970), 53-71.
Ono, H., ‘Kripke models and intermediate logics’ Publications of Research Institute for Mathematical Sciences, Kyoto University 6 (1970), 461-476.
Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, PWN, Warszawa, 1963.
Scroggs, S. G., ‘Extensions of the Lewis system S5’ Journal of Symbolic Logic 16 (1951), 111-120.
Sikorski, R., Boolean Algebras, Springer Verlag, Berlin, Goettingen, Heidelberg, New York, 1964.
Suzuki, N.-Y., ‘An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics’ Studia Logica 48 (1989), 141-155.
Troelstra, A. S., ‘On intermediate propositional logics’ Indagationes Mathematicae 27 (1965), 141-152.
Rights and permissions
About this article
Cite this article
Bezhanishvili, G. Varieties of Monadic Heyting Algebras. Part III. Studia Logica 64, 215–256 (2000). https://doi.org/10.1023/A:1005285631357
Issue Date:
DOI: https://doi.org/10.1023/A:1005285631357