Hostname: page-component-6766d58669-h8lrw Total loading time: 0 Render date: 2026-05-15T03:14:02.666Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ALGEBRAIC APPROACH TO CANONICAL FORMULAS: INTUITIONISTIC CASE

Published online by Cambridge University Press:  05 October 2009

GURAM BEZHANISHVILI  and
NICK BEZHANISHVILI
GURAM BEZHANISHVILI*
Affiliation:
Department of Mathematical Sciences, New Mexico State University
NICK BEZHANISHVILI*
Affiliation:
Department of Computing, Imperial College London
*
*DEPARTMENT OF MATHEMATICAL SCIENCES, NEW MEXICO STATE UNIVERSITY, LAS CRUCES, NM 88003 E-mail: gbezhani@nmsu.edu
†DEPARTMENT OF COMPUTING, IMPERIAL COLLEGE LONDON, 180 QUEEN’S GATE, LONDON SW7 2AZ, UK E-mail: nbezhani@doc.ic.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧, →) homomorphisms, (∧, → , 0) homomorphisms, and (∧ , → , ∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s theorem that each intermediate logic can be axiomatized by canonical formulas.

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 2 , Issue 3 , September 2009 , pp. 517 - 549
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

BIBLIOGRAPHY

Bezhanishvili, G., & Bezhanishvili, N. (2008). Profinite Heyting algebras. Order, 25(3), 211–227.CrossRefGoogle Scholar
Bezhanishvili, G., & Ghilardi, S. (2007). An algebraic approach to subframe logics. Intuitionistic case. Annals of Pure and Applied Logic, 147(1–2), 84–100.CrossRefGoogle Scholar
Bezhanishvili, G., & Jansana, R. (2008). Duality for Distributive and Implicative Semi-Lattices. Submitted. Available from:http://www.mat.ub.edu/logica/docs/BeJa08-m.pdf.Google Scholar
Bezhanishvili, N. (2006). Lattices of intermediate and cylindric modal logics. PhD Thesis, University of Amsterdam.Google Scholar
Bezhanishvili, N. (2008). Frame based formulas for intermediate logics. Studia Logica, 90, 139–159.CrossRefGoogle Scholar
Blok, W. (1976). Varieties of interior algebras. PhD Thesis, University of Amsterdam.Google Scholar
Blok, W. (1978). On the degree of incompleteness of modal logics. Polish Academy of Sciences Institute of Philosophy and Sociology Bulletin of the Section of Logic, 7(4), 167–175.Google Scholar
Blok, W. (1980). The lattice of modal logics: An algebraic investigation. Journal of Symbolic Logic, 45(2), 221–236.CrossRefGoogle Scholar
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic. Oxford Logic Guides, Vol. 35. New York, NY: The Clarendon Press Oxford University Press.CrossRefGoogle Scholar
De Jongh, D. (1968). Investigations on the intuitionistic propositional calculus. PhD Thesis, University of Wisconsin.Google Scholar
Diego, A. (1966). Sur les Algèbres de Hilbert. Translated from the Spanish by Luisa, Iturrioz. Collection de Logique Mathématique, Sér. A, Fasc. XXI. Paris, France: Gauthier-Villars.Google Scholar
Esakia, L. (1974). Topological Kripke models. Soviet Mathematics Doklady, 15, 147–151.Google Scholar
Esakia, L. (1979). On the theory of modal and superintuitionistic systems. In Logical Inference (Moscow, 1974). Moscow, Russian Federation: Nauka.Google Scholar
Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Tbilisi, Georgia: Metsniereba.Google Scholar
Fine, K. (1974a). An ascending chain of S4 logics. Theoria, 40(2), 110–116.CrossRefGoogle Scholar
Fine, K. (1974b). An incomplete logic containing S4. Theoria, 40(1), 23–29.CrossRefGoogle Scholar
Fine, K. (1985). Logics containing K4. II. Journal of Symbolic Logic, 50(3), 619–651.CrossRefGoogle Scholar
Jankov, V. (1963). On the relation between deducibility in intuitionistic propositional calculus and finite implicative structures. Doklady Akademii Nauk SSSR, 151, 1293–1294.Google Scholar
Jankov, V. (1968). The construction of a sequence of strongly independent superintuitionistic propositional calculi. SovietMathematics Doklady, 9, 806–807.Google Scholar
Jónsson, B. (1967). Algebras whose congruence lattices are distributive. Mathematica Scandinavica, 21, 110–121 (1968).CrossRefGoogle Scholar
Köhler, P. (1981). Brouwerian semilattices. Transactions of the American Mathematical Society, 268(1), 103–126.CrossRefGoogle Scholar
Litak, T. (2002). A continuum of incomplete intermediate logics. Reports on Mathematical Logic, 36, 131–141.Google Scholar
McKay, C. (1968). The decidability of certain intermediate propositional logics. Journal of Symbolic Logic, 33, 258–264.CrossRefGoogle Scholar
McKenzie, R. (1972). Equational bases and nonmodular lattice varieties. Transactions of the American Mathematical Society, 174, 1–43.CrossRefGoogle Scholar
Minari, P. (1986). Intermediate logics with the same disjunctionless fragment as intuitionistic logic. Studia Logica, 45(2), 207–222.CrossRefGoogle Scholar
Nemitz, W. (1965). Implicative semi-lattices. Transactions of the American Mathematical Society, 117, 128–142.CrossRefGoogle Scholar
Nishimura, I. (1960). On formulas of one variable in intuitionistic propositional calculus. Journal of Symbolic Logic, 25, 327–331.CrossRefGoogle Scholar
Rasiowa, H., & Sikorski, R. (1963). The Mathematics of Metamathematics. Monografie Matematyczne, Tom 41. Warsaw, Poland: Państwowe Wydawnictwo Naukowe.Google Scholar
Rieger, L. (1949). On the lattice theory of Brouwerian propositional logic. Acta Facultatis Rerum Naturalium Universitatis Carolinae, 189, 1–40.Google Scholar
Shehtman, V. (1977). Incomplete propositional logics. Doklady Akademii Nauk SSSR, 235(3), 542–545.Google Scholar
Tomaszewski, E. (2003). On sufficiently rich sets of formulas. PhD Thesis, Institute of Philosophy, Jagellonian University, Krakóv.Google Scholar
Umezawa, T. (1959). On intermediate propositional logics. Journal of Symbolic Logic, 24, 20–36.CrossRefGoogle Scholar
Visser, A., de Jongh, D., van Benthem, J., & Renardel de Lavalette, G. (1995). NNIL a study in intuitionistic logic. In Ponse, A., de Rijke, M., and Venema, Y., editors. Modal Logics and Process Algebra: A Bisimulation Perspective. pp. 289–326.Google Scholar
Willard, S. (1970). General Topology. Reading, MA, London, Don Mills, Ontario: Addison-Wesley Publishing Co.Google Scholar
Wronski, A. (1973). Intermediate logics and the disjunction property. Reports on Mathematical Logic, 1, 39–51.Google Scholar
Zakharyaschev, M. (1983). On intermediate logics. Soviet Mathematics Doklady, 27(2), 274–277.Google Scholar
Zakharyaschev, M. (1987). The disjunction property of superintuitionistic and modal logics. Matematicheskie Zametki, 42(5), 729–738, 763.Google Scholar
Zakharyaschev, M. (1989). Syntax and semantics of superintuitionistic logics. Algebra and Logic, 28(4), 262–282.CrossRefGoogle Scholar
Zakharyaschev, M. (1992). Canonical formulas for K4. I. Basic results. Journal of Symbolic Logic, 57(4), 1377–1402.CrossRefGoogle Scholar
Zakharyaschev, M. (1996). Canonical formulas for K4. II. Cofinal subframe logics. Journal of Symbolic Logic, 61(2), 421–449.CrossRefGoogle Scholar
Zakharyaschev, M. (1997). Canonical formulas for modal and superintuitionistic logics: A short outline. In Advances in Intensional Logic, Applied Logic Series, Vol. 7. Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar