# Codimension and pseudometric in co-Heyting algebras

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 $d$.
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Article dans une revue
Algebra Universalis, Springer Verlag, 2010, 64 (3-4), pp.251-282
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https://hal.archives-ouvertes.fr/hal-00346038
Contributeur : Luck Darnière <>
Soumis le : mercredi 10 décembre 2008 - 21:37:10
Dernière modification le : mercredi 10 octobre 2018 - 19:56:03
Document(s) archivé(s) le : mardi 8 juin 2010 - 16:09:54

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• HAL Id : hal-00346038, version 1
• ARXIV : 0812.2026

### Citation

Luck Darnière, Markus Junker. Codimension and pseudometric in co-Heyting algebras. Algebra Universalis, Springer Verlag, 2010, 64 (3-4), pp.251-282. 〈hal-00346038〉

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