Submission date: 18 Decembre 2008.

Abstract:

In this paper we examine dimension and codimension in co-Heyting algebras. Codimension gives rise to a pseudometric on a co-Heyting algebra L. 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 coincides with the profinite completion of L if and only if it is compact. 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. 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.

Mathematics Subject Classification: 06D20, 06B23, 06B30, 06D50

Keywords and phrases: Heyting algebra, co-Heyting algebra, Brouwerian lattice, dimension, codimension, slice, completion, precompact Heyting algebra, finitely generated Heyting algebra

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