Subprojective lattices and projective geometry

Abstract

The class of lattices we are interested in (subprojective lattices), can be gotten by taking the MacNeille completions of the class of complemented, modular, atomic lattices. McLaughlin showed that subprojective lattices can be represented as the lattices of W-closed subspaces of a vector space U in duality with a vector space W. In this paper, we give a characterization of subprojective lattices in terms of atoms and dual atoms, by means of an incidence space satisfying self-dual axioms. In the finite-dimensional case, a subprojective lattice is projective, and hence our self-dual axioms characterize finite-dimensional projective spaces in terms of points and hyperplanes. No numerical parameters appear explicitly in these axioms. For each subprojective lattice L with at least three elements, we define a projective envelope P ( L) for it. P ( L) is a projective lattice and there is a natural inf-preserving injection of L into P ( L). This injection has other important properties which we take as the definition of a geometric map. In the course of studying geometric maps, we obtain a lattice theoretic proof of Mackey's result that the join of a U-closed subspace of V and a finite-dimensional subspace is U-closed, where ( U, V) form a dual pair of vector spaces over a division ring. Furthermore, we show that if L is a subprojective lattice, P a projective lattice, and ψ: L → P a geometric map, then P is isomorphic to the projective envelope P ( L) of L. The paper presents many other properties of subprojective lattices. It concludes with a characterization of subprojective lattices which are also projective.