Abstract
We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding to an introduction rule and the other to an elimination rule, thus providing a manageable way to deal with these operations. Our main result is negative: all operations that arise turn out to be Heyting terms or the mentioned already known operations or operations interdefinable with them. However, it should be noted that some of the operations that arise may exist even if the known operations do not. We also study the extension of Priestley duality to Heyting algebras enriched with the new operations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balbes, R., and P. Dwinger, Distributive Lattices, University of Missouri Press, 1974.
Belnap N.: ‘Tonk, Plonk and Plink’. Analysis 22, 130–134 (1962)
Caicedo X., Cignoli R.: ‘An algebraic approach to intuitionistic connectives’. Journal of Symbolic Logic 66, 1620–1636 (2001)
Castiglioni J.L., Sagastume M., San Martín H.J.: ‘Frontal Heyting algebras’. Reports on Mathematical Logic 45, 201–224 (2010)
Castiglioni J.L., San Martín H.J.: ‘On the variety of Heyting algebras with successor generated by finite chains’. Reports on Mathematical Logic 45, 225–248 (2010)
Ertola Biraben R.C.: ‘On some operations using the min operator’. in Studies in Logic 21, 353–368 (2009) College Publications, London
Esakia L.: ‘The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic’. Journal of Applied Non-Classical Logics 16, 349–366 (2006)
Esakia L.: ‘Topological Kripke models’. Soviet Math. Dokl. 15, 147–151 (1974)
Esakia L., ‘On the theory of modal and superintuitionistic systems’. In Logical Inference, Nauka, Moscow, 1979, pp. 147–172.
Esakia L., ‘Heyting Algebras I. Duality Theory’ (Russian). Metsniereba, Tbilisi, 1985.
Gabbay D.: ‘On some new intuitionistic connectives, I’. Studia Logica 33, 127–139 (1977)
Grätzer G.: ‘On boolean functions (Notes on lattice theory II)’. Revue Roumaine de Mathmatiques Pures et Apliquees 7, 693–697 (1962)
Kaarly, K., and A. Pixley, Polynomial completeness in algebraic systems, Chapman and Hall, 2000.
Kuznetsov A.: ‘On the propositional calculus of intuitionistic provability’. Soviet Math. Dokl. 32, 18–21 (1985)
Pixley A.: ‘Completeness in arithmetical algebras’. Algebra Universalis 2, 179–196 (1972)
Smetanich Y.: ‘On the Completeness of a Propositional Calculus with a Supplementary Operation in one Variable’. Tr. Mosk. Mat. Obsch. 9, 357–371 (1960)
Yashin A.: ‘The Smetanich logic T Φ and two definitions of a new intuitionistic connective’. Mathematical Notes 56, 745–750 (1994)
Yashin A.: ‘New solutions to Novikov’s problem for intuitionistic connectives’. J. Logic Computat. 8, 637–664 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ertola Biraben, R.C., San Martín, H.J. On Some Compatible Operations on Heyting Algebras. Stud Logica 98, 331–345 (2011). https://doi.org/10.1007/s11225-011-9338-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-011-9338-y