On Some Compatible Operations on Heyting Algebras

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.Facultad de Ciencias ExactasFacultad de Humanidades y Ciencias de la Educació

On Principal Congruences in Distributive Lattices with a Commutative Monoidal Operation and an Implication

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.Fil: Jansana Ferrer, Ramon. Universidad de Barcelona; EspañaFil: San Martín, Hernán Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin

On Some Compatible Operations on Heyting Algebras

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.Fil: Ertola Biraben, Rodolfo Cristian. Universidad Nacional de La Plata. Facultad de Humanidades y Ciencias de la Educación; Argentina

On Some Compatible Operations on Heyting Algebras

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.Fil: Ertola Biraben, Rodolfo Cristian. Universidad Nacional de La Plata. Facultad de Humanidades y Ciencias de la Educación; Argentina

On Some Compatible Operations on Heyting Algebras

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.Fil: Ertola Biraben, Rodolfo Cristian. Universidad Nacional de La Plata. Facultad de Humanidades y Ciencias de la Educación; Argentina

On Some Extensions Of Intuitionistic Logic

We prove that many extensions of Intuitionistic Sentential Calculus ISC with new intuitionistic connectives that are known to be conservative extensions of ISC are not conservative extensions of Intuitionistic Predicate Calculus because formulas such as Kuroda's are derivable. We thus solve a problem posed by López-Escobar in 1985 and answer a question posed by Humberstone in 2001 regarding a connective called the strongest anticipator.411-21722Caicedo, X., Cignoli, R., An algebraic approach to intuitionistic connectives (2001) The Journal of Symbolic Logic, 66, pp. 1620-1636Ertola Biraben, R.C., San Martín, H.J., On some compatible operations on heyting algebras (2011) Studia Logica, 98, pp. 331-345Esakia, L., Quantification in intuitionistic logic with provability smack (1998) Bulletin of the Section of Logic, 27, pp. 26-28Esakia, L., Scattered toposes (2000) Annals of Pure and Applied Logic, 103, pp. 97-107Esakia, L., The modalized heyting calculus: A conservative modal extension of the intuitionistic logic (2006) Journal of Applied Non-Classical Logics, 16, pp. 349-366Gabbay, D.M., On some new intuitionistic propositional connectives. I (1977) Studia Logica, 36, pp. 127-139Gabbay, D.M., (1981) Semantical Investigations in Heyting's Intuitionistic Logic, , Reidel Publishing Company, DordrechtHumberstone, L., The pleasures of anticipation: Enriching intuitionistic logic (2001) Journal of Philosophical Logic, 30, pp. 395-438Humberstone, L., (2011) The Connectives, , MIT Press, Cambridge, MassKuznetsov, A., On the propositional calculus of intuitionistic provability (1985) Soviet Math. Dokl., 32, pp. 18-21López-Escobar, E.G.K., On intuitionistic sentential connectives I (1985) Revista Colombiana de Matemáticas, 19, pp. 117-130Smetanich, Y., On the completeness of a propositional calculus with a supplementary operation in one variable (1960) Tr. Mosk. Mat. Obsch., 9, pp. 357-371. , (in Russian)(1962) MR, 24, pp. A680. , reviewedSmetanich, Y., On propositional calculi with an additional connective (1961) Soviet Math. Doklady, 139, pp. 309-312. , (in Russian)(1963) MR, 26. , reviewedTroelstra, A., Van Dalen, D., (1988) Constructivism in Mathematics. An Introduction, 1. , North-Holland, AmsterdamYashin, A.D., New solutions for novikov's problem for intuitionistic connectives (1998) Journal of Logic and Computation, 8, pp. 637-66