# Codimension and pseudometric in co-Heyting algebras

- 45 Downloads

## Abstract

In this paper, we introduce a notion of dimension and codimension for every element of a bounded distributive 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*.

## 2010 Mathematics Subject Classification

Primary: 06D20 Secondary: 06B23 06B30 06D50## Keywords and phrases

Lattice theory Heyting algebras dimension codimension slices duality general topology## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bellissima F.: Finitely generated free Heyting algebras. J. Symbolic Logic
**51**, 152–165 (1986)CrossRefMATHMathSciNetGoogle Scholar - 2.Bezhanishvili G., Grigolia R.: Locally finite varieties of Heyting algebras. Algebra Universalis
**54**, 465–473 (2005)CrossRefMATHMathSciNetGoogle Scholar - 3.Bezhanishvili G., Grigolia R., Gehrke M., Mines R., Morandi P.J.: Profinite completions and canonical extensions of Heyting algebras. Order
**23**, 143–161 (2006)CrossRefMATHMathSciNetGoogle Scholar - 4.Darnière, L., Junker, M.: On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond. Preprint (2008). http://arxiv.org/archive/math
- 5.Fitting M.C.: Intuitionistic logic, model theory and forcing. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1969)Google Scholar
- 6.Ghilardi, S.: Unification in intuitionistic logic. In: J. Symbolic Logic
**64**, 859–880 (1999)Google Scholar - 7.Hochster M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc.
**142**, 43–60 (1969)CrossRefMATHMathSciNetGoogle Scholar - 8.Hosoi T.: On intermediate logics. I. J. Fac. Sci. Univ. Tokyo Sect. I
**14**, 293–312 (1967)MathSciNetGoogle Scholar - 9.Komori Y.: The finite model property of the intermediate propositional logics on finite slices. J. Fac. Sci. Univ. Tokyo Sect. IA Math.
**22**, 117–120 (1975)MATHMathSciNetGoogle Scholar - 10.Kuznetsov, A.V.: On superintuitionistic logics. In: Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), vol. 1, pp. 243–249. Canad. Math. Congress, Montreal, Quebec (1975)Google Scholar
- 11.Mardaev S.I.: The number of prelocal-tabular superintuitionistic propositional logics. Algebra and Logic
**23**, 56–66 (1984)CrossRefMATHMathSciNetGoogle Scholar - 12.Ono, H.: Kripke models and intermediate logics. Publ. Res. Inst. Math. Sci.
**6**, 461–476 (1970/71)Google Scholar - 13.Popkorn S.: First steps in modal logic. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar