Errata on “On the Variety of Heyting Algebras with Successor Generated by all Finite Chains”

Errata on “On the Variety of Heyting Algebras with Successor Generated by all Finite Chains

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 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!.

Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, and G operations as well as expansions of some commutative integral residuated lattices with successor operations

Codimension and pseudometric in co-Heyting algebras

In this paper we introduce a notion of dimension and codimension for every element of a distributive bounded lattice $L$. These notions prove to have a good behavior when $L$ is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on $L$ which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of $L$ with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of $L$ if and only if it is compact or equivalently if every finite dimensional quotient of $L$ is finite. In this case we say that $L$ is precompact. If $L$ is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers $n,d$ of a term $t_{n,d}$ such that in every co-Heyting algebra generated by an $n$-tuple $a$, $t_{n,d}(a)$ is precisely the maximal element of codimension $d$.Comment: 34 page

Bi-intermediate logics of trees and co-trees

A bi-Heyting algebra validates the G\"odel-Dummett axiom $(p\to q)\vee (q\to p)$ iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-G\"odel algebras and form a variety that algebraizes the extension $\mathsf{bi}$-$\mathsf{LC}$ of bi-intuitionistic logic axiomatized by the G\"odel-Dummett axiom. In this paper we initiate the study of the lattice $\Lambda(\mathsf{bi}$-$\mathsf{LC})$ of extensions of $\mathsf{bi}$-$\mathsf{LC}$. We develop the methods of Jankov-style formulas for bi-G\"odel algebras and use them to prove that there are exactly continuum many extensions of $\mathsf{bi}$-$\mathsf{LC}$. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of $\mathsf{bi}$-$\mathsf{LC}$. We introduce a sequence of co-trees, called the finite combs, and show that a logic in $\mathsf{bi}$-$\mathsf{LC}$ is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$ and consequently, a unique pre-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$. These results contrast with the case of the intermediate logic axiomatized by the G\"odel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular

Frontal Operators in Weak Heyting Algebras

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation τ(a) ≤ b ∨ (b → a), for all a, b ∈ A. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces hX, ≤, T, Ri where hX, ≤, T i is a WH - space [6], and R is an additional binary relation used to interpret the modal operator. We will also study the WH -algebras with successor and the WH -algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH -spaces.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; ArgentinaFil: San Martín, Hernán Javier. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

On locally finite varieties of Heyting algebras

For every $n \in \mathbb{N}$, we construct a variety of Heyting algebras, whose $n$-generated free algebra is finite but whose $(n+1)$-generated free algebra is infinite