On the variety of Heyting algebras with successor generated by all finite chains

Contrary to the variety of Heyting algebras, finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor. In particular, all finite chains generate a proper subvariety, SLHω, of the latter. There is a categorical duality between Heyting algebras with successor and certain Priestley spaces. Let X be the Heyting space associated by this duality to the Heyting algebra with successor H. If there is an ordinal κ and a filtration on X such that X = S λ≤κ Xλ, the height of X is the minimun ordinal ξ ≤ κ such that Xc ξ = ∅. In this case, we also say that H has height ξ. This filtration allows us to write the space X as a disjoint union of antichains. We may think that these antichains define levels on this space. We study the way of characterize subalgebras and homomorphic images in finite Heyting algebras with successor by means of their Priestley spaces. We also depict the spaces associated to the free algebras in various subcategories of SLH.Fil: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: San Martín, Hernán Javier. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin

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 the Variety of Heyting Algebras with Successor Generated by All Finite Chains

Contrary to the variety of Heyting algebras, finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor. In particular, all finite chains generate a proper subvariety, SLH!, of the latter. There is a categorical duality between Heyting algebras with successor and certain Priestley spaces. Let X be the Heyting space associated by this duality to the Heyting algebra with successor H. If there is an ordinal and a filtration S on X such that X = X, the height of X is the minimun ordinal ≤ such that Xc = ∅. In this case, we also say that H has height . This filtration allows us to write the space X as a disjoint union of antichains. We may think that these antichains define levels on this space. We study the way of characterize subalgebras and homomorphic images in finite Heyting algebras with successor by means of their Priestley spaces. We also depict the spaces associated to the free algebras in various subcategories of SLH!.

On Frontal Heyting Algebras

A frontal operator in a Heyting algebra is an expansive operator preserving finite meets which also satisfies the equation \tau(\x) \le y \vee (y\rightarrow x). A frontal Heyting algebra is a pair $(H,\tau)$, where $H$ is a Heyting algebra and $\tau$ 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 $\gamma$, (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