A dimension theory fo (dual) Heyting algebras
by
Luck Darniere
Dept. Mathematiques, Univ. Angers (France)
Coauthors: Markus Junker, Univ. Freiburg (Germany)

The height of a prime filter p of a distributive bounded lattice L is the maximal length of an ascending chain of prime filters contained in p . We denote by zero the smallest element of L. The codimension of a nonzero element of L is the minimum of the heights of the prime filters of L which contain this element. The codimension of zero is minus infinity. These definitions come from classical algebraic geometry. We expect them to give new insight in the theory of (dual) Heyting algebras.

Exemple 0: Let L be the lattice of all closed subsets of an algebraic variety X over an algebraically closed field. The dual of L (that is L with reverse order) is a Heyting algebra. For any element A of L, the above defined codimension of A in L coincides with the usual geometric notion of codimension of A in X.

Given a lattice L whose dual is a Heyting algebra we denote by nL the set of elements of L whose codimension is greater than or equal to n. This is an ideal of L.

Theorem 1. Let L be a lattice whose dual is a finitely generated Heyting algebra. Then for any positive integer n the quotient L/nL is finite.

Theorem 2. Let L be a lattice whose dual is a finitely presented Heyting algebra. Then every nonzero element of L has finite codimension; d(A, B)=2^{-codim A * B}, where A * B denotes the symetric difference operation in L, is an ultrametric distance on L; the completion of L with respect to this distance is also the projective limit of all the finite quotients L/nL (as n ranges over all the positive integers).

The class of dual Heyting algebras of fixed finite dimension is elementary. For each positive integer N we introduce a very large class of N-dimensional dual Heyting algebras, we call N-scaled lattices, which contains every geometric example such as example 0.

Theorem 3. The theory of N-scaled lattices admits a model-companion, which eliminates the quantifier in a finite extension by definition of the lattice language.

We explicitly axiomatize this model-companion by two algebraic properties, we call catenarity and splitting (Arxiv maths.LO/0606792). It is known from a theorem of A. Pitts (JSL 57(1):23-52, 1992) rephrased in model-theoretic terms by S. Ghilardi and M. Zawadowzki (APAL 88:27-46, 1997), that the theory of all Heyting algebras admits a model-completion. However no axiomatisation is known yet for this theory. Perhaps the following observation will change this situation in the future:

Theorem 4. The dual of any existentially closed Heyting algebra satisfies the splitting property.

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Date received: April 28, 2007