Codimension and pseudometric in co-Heyting algebras

TL;DR

This paper introduces a dimension and codimension framework for distributive bounded lattices, especially co-Heyting algebras, establishing a pseudometric structure and analyzing their completions, with applications to algebraic generation and element characterization.

Contribution

It defines a codimension-based pseudometric on co-Heyting algebras and characterizes their completions, linking algebraic properties with metric and topological structures.

Findings

The pseudometric satisfies the ultrametric inequality.

The Hausdorff completion equals the projective limit of finite dimensional quotients.

Finitely presented co-Heyting algebras are precompact Hausdorff.

Abstract

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 $d$.