Abstract
Finitely generated Heyting algebras and monadic Heyting algebras are described in terms of perfect Kripke models using colouring technique. Defining projective algebras as retract of free algebras, the characteristic of finitely generated projective algebras is given in varieties of Heyting algebras and monadic Heyting algebras. By means of projective algebras, using an algebraic proof, the conditions of Friedman’s conjecture (Problem 41 [8]) are confirmed for well-known Medvedev's logic.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
R. Balbs and A. Horn, Injective and projective Heyting algebra, Trans. Amer. Soc. 148 (1970), 549–559.
R.Bull, MPS as the formalization of an intuitionistic concept of modality, Journal of Symbolic Logic 31 (1966), 609–616.
R. Engelking, General topology, Polish Scientific Publishers, Warszawa, 1977.
L. Esakia, Topological Kripke model, Dokl. Sov. Akad. 214, 2, 298–301.
L. Esakia, The provability status of intuitionistic logic with the maximality principle, Preprint, 1989.
L. Esakia, The provability logic with quantifier modalities, IYSoviet-Finnish Symposium on Intensional Logic and Logical Structure of Theory "Metsniereba", Tbilisi, (1988), 4–10.
L. Esakia, R. Grigolia, The criterion of Brouwerian and closure algebras to be finiteliy generated, Bull. Sect. Logic 6, 2, (1973), 46–52.
H. Friedman, One hundred and two problems in mathematical logic, Journal of Symbolic Logic 40 (1975), 113–129.
R. Grigolia, Free algebras of non classical logics, Monograph, “Metsniereba”, Tbilisi, 1987.
P. Halmos, Algebraic logic, Chesea Publ. Comp., N. Y., 1962.
R. Harrop, Concerning formulas of types A ⇀ B V C, A⇀ (∃x)B(x) in intuitionistic formal system, Journal of Symbolic Logic 25 (1960), 27–32.
G. Kreisel and H. Putnam, Eine Unableitbarkeitsbeweismethode für den ituitionistischen Aussagenkalkul, Arch. Math. Logic (Archiv für Math. Logik und Grundlagenforsch.) 3 (1957), 74–78.
JV. Medvedev, Finite problems, Dokl. Sov. Acad. 142, 5 (1962), 1015–1018.
I. Nishimura, On formulas on one variable in intuitionistic propositional calculus, Journal of Symbolic Logic 25 (1960), 327–331.
H. Ono, On some intuitionistic modal logic, Publication of the Research Institute for mathematical sciences, Kyoto university, 13 (1977), 687–722.
W. A. Pogorzelski, Structural completeness of the propositional calculus, Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. 19 (1971), 349–351.
T. Prucnal, Structural completeness of Medvedev's propositional calculus, Reports on Math. Log. 6 (1976).
T. Prucnal, On two problems of Harvey Friedman, SL 38, 3 (1979), 247–262.
T. Prucnal, On the structural completeness of some pure implicational propositional calculus, SL 30 (1972), 45–52.
H. Rasiowa and R. Sikorski, The mathematics of metamathematics, (3rd ed.) Monografie mathematyczne, Warszawa, 1970.
L. Rieger, Zametki o tak nazyvaemykh algebrakh s zamykaniami, Chekh. Math. Journ. 7, 16 (1957).
N. I. Suzuki, An algrbraic approach to intuitionistic modal logic in connections with Intermediate predicate logic, SL 48 (1989), 141–155.
A. Urquhart, Free Heyting algebras, Alg. Univ. 3, 1, (1973), 94–97.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Grigolia, R. (1995). Free and projective Heyting and monadic Heyting algebras. In: Höhle, U., Klement, E.P. (eds) Non-Classical Logics and their Applications to Fuzzy Subsets. Theory and Decision Library, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0215-5_4
Download citation
DOI: https://doi.org/10.1007/978-94-011-0215-5_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4096-9
Online ISBN: 978-94-011-0215-5
eBook Packages: Springer Book Archive