On Frontal Heyting Algebras

José L. Castiglioni,

Marta S. Sagastume,

Hernán J. San Martín


A frontal operator in a Heyting algebra is an expansive operator preserving finite meets which also satisfies the equation (x) ≤ y ∨ (y → x). A frontal Heyting algebra is a pair (H, ), where H is a Heyting algebra and  a frontal operator on H. Frontal operators are always compatible, but not necessarily new or implicit in the sense of Caicedo and Cignoli (An algebraic approach to intuitionistic connectives. Journal of Symbolic Logic, 66, No4 (2001), 1620-1636). Classical examples of new implicit frontal operators are the functions, (op. cit., Example 3.1), the successor (op. cit., Example 5.2), and Gabbay’s operation (op. cit., Example 5.3).
We study a Priestley duality for the category of frontal Heyting algebras and in particular for the varieties of Heyting algebras with each one of the implicit operations given as examples. The topological approach of the compatibility of operators seems to be important in the research of affin completeness of Heyting algebras with additional compatible operations. This problem have also a logical point of view. In fact, we look for some complete propositional intuitionistic calculus enriched with implicit connectives.

[1] R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, Miss. (1974).
[2] X. Caicedo and R. Cignoli, An algebraic approach to intuitionistic connectives, Journal of Symbolic Logic 66, No.4 (2001), pp. 1620–1636.
[3] S. Celani, Distributive lattices with a negation operator, Mathematical Logic Quarterly 45 (1999), pp. 207–218.
[4] S. Celani and R. Jansana, Bounded distributive lattices with strict implication, Mathematical Logic Quarterly 51 (2005), pp. 219–246.
[5] R. Cignoli, S. Lafalce and A. Petrovich, Remarks on Priestley duality for distributive lattices, Orden 8 (1991), pp. 183–197.
[6] L. Esakia, The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic, Journal of Applied Non-Classical Logics 16, No.3-4 (2006), pp. 349–366.
[7] D. M. Gabbay, On some new intuitionistic propositional connectives. I, Studia Logica 36 (1977), pp. 127–139.
[8] P. Jonstone, Stone Spaces. Cambridge University Press, 1982.
[9] A. V. Kusnetsov, On the Propositional Calculus of Intuitionistic Provability, Soviet Math. Dokl. 32 (1985), pp. 18–21.
[10] P. Morandi, Dualities in Lattice Theory, Mathematical Notes http://sierra.nmsu.edu/morandi/.
[11] E. Orlowska and I. Rewitzky, Discrete Dualities for Heyting algebras with Operators, Fundamenta Informaticae 81 (2007), pp. 275–295.
[12] A. D. Yashin, New solutions to Novikov’s problem for intuitionistic connectives, Journal of Logic and Computation 8 (1998), pp. 637–664.

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.