1 research outputs found
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 . These notions prove to have a
good behavior when is a co-Heyting algebra. In this case the codimension
gives rise to a pseudometric on which satisfies the ultrametric triangle
inequality. We prove that the Hausdorff completion of 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 if and only if it is compact or equivalently if
every finite dimensional quotient of is finite. In this case we say that
is precompact. If 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 of a term
such that in every co-Heyting algebra generated by an -tuple ,
is precisely the maximal element of codimension .Comment: 34 page