Lattices generated by subspaces in d-bounded distance-regular graphs

Let Gamma denote a d-bounded distance-regular graph with diameter d >= 2. A regular strongly closed subgraph of Gamma is said to be a subspace of Gamma Define the empty set empty set to be the subspace with diameter -1 in Gamma. For 0 <= i <= i + s <= d - 1, let L(i, i + s) denote the set of all subspaces in Gamma with diameters i, i + 1, ..., i + s including Gamma and empty set. If we define the partial order on L(i, i + s) by ordinary inclusion (resp. reverse inclusion), then L(i, i + s) is a poset, denoted by L-O(i, i + s) (resp. L-R(i, i + s)). In the present paper we show that both L-O(i, i + s) and L-R (i, i + s) are atomic lattices, and classify their geometricity. (C) 2007 Elsevier B.V. All rights reserved.