Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators

Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

Ruitenburg\u2019s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately periodic if f fixes all the generators but one. More precisely, there is N 65 0 such that f^N+2 = f^N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators

Colimits of Heyting Algebras through Esakia Duality

In this note we generalize the construction, due to Ghilardi, of the free Heyting algebra generated by a finite distributive lattice, to the case of arbitrary distributive lattices. Categorically, this provides an explicit construction of a left adjoint to the inclusion of Heyting algebras in the category of distributive lattices This is shown to have several applications, both old and new, in the study of Heyting algebras: (1) it allows a more concrete description of colimits of Heyting algebras, as well as, via duality theory, limits of Esakia spaces, by knowing their description over distributive lattices and Priestley spaces; (2) It allows a direct proof of the amalgamation property for Heyting algebras, and of related facts; (3) it allows a proof of the fact that the category of Heyting algebras is co-distributive. We also study some generalizations and variations of this construction to different settings. First, we analyse some subvarieties of Heyting algebras -- such as Boolean algebras, $\mathsf{KC}$ and $\mathsf{LC}$ algebras, and show how the construction can be adapted to this setting. Second, we study the relationship between the category of image-finite posets with p-morphisms and the category of posets with monotone maps, showing that a variation of the above ideas provides us with an appropriate general idea.Comment: 27 page

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