A Geometric Description of Modular Lattices

Baer [1] observed that modular lattices of finite length (for example subgroup lattices of abelian groups) can be conceived as subspace lattices of a projective geometry structure on an ordered point set; the set of join irreducibles which in this case are the cyclic subgroups of prime power order. That modular lattices of finite length can be recaptured from the order on the points and, in addition, the incidence of points with ‘lines’, the joins of two points, or the blocks of collinear points has been elaborated by Kurinnoi [18] , Faigle and Herrmann [7], Benson and Conway [2] , and , in the general framework of the ‘core’ of a lattice, by Duquenne [5]. In [7] an axiomatization in terms of point-line incidence has been given. Here, we consider, more generally, modular lattices in which every element is the join of completely join irreducible ‘points’. We prove the isomorphy of an algebraic lattice of this kind and the associated subspace lattice and give a first order characterization of the associated ‘ordered spaces’ in terms of collinearity and order which appears more natural and powerful. The crucial axioms are a ‘triangle axiom’, which includes the degenerate cases, and a strengthened ‘line regularity axiom’, both derived from [7]. As a consequence, using Skolemization, we get that any variety of modular lattices is generated by subspace latices of countable spaces. The central concept, connecting the geometric structure and the lattice structure, is that of a line interval (p + q)/(p + q) where p and q are points and p, q their unique lower covers. Given a line interval, any choice of one incident point per atom of the line interval produces a line of the space.