Locally Polar Lattices

A lattice 9 is locally polar if each element is a join of points (atoms) and each interval L?$ = (y j x < y}, x a point, is a polar space (see Tits [6] or Buekenhout and Shult [4]). Recall that the rank of a polar space is the maximum projective dimension of an element; since this number is finite, 9 has a 1 and a 0. There is an obvious dimension function on 9, so lines and planes have the obvious meaning. We assume that Y has the following properties.