Codimension and pseudometric on co-Heyting algebras

by L. Darnière and M. Junker

Algebra Universalis 64 (2010) no. 3-4, 251-282

Submitted on March 19, 2009.

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.

Mathematics Subject Classification

06D20 Heyting algebras [See also 03G25]
06B23 Complete lattices, completions
06B30 Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12]
06D50 Lattices and duality

Electronic version of the paper

Version December 2008 (34 pages) pdf