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

Let ΓΓ denote a d  -bounded distance-regular graph with diameter d⩾2d⩾2. A regular strongly closed subgraph of ΓΓ is said to be a subspace of ΓΓ. Define the empty set ∅∅ to be the subspace with diameter -1-1 in ΓΓ. For 0⩽i⩽i+s⩽d-10⩽i⩽i+s⩽d-1, let L(i,i+s) denote the set of all subspaces in ΓΓ with diameters i,i+1,…,i+si,i+1,…,i+s including ΓΓ and ∅∅. If we define the partial order on L(i,i+s)L(i,i+s) by ordinary inclusion (resp. reverse inclusion), then L(i,i+s)L(i,i+s) is a poset, denoted by LO(i,i+s)LO(i,i+s) (resp. LR(i,i+s)LR(i,i+s)). In the present paper we show that both LO(i,i+s)LO(i,i+s) and LR(i,i+s)LR(i,i+s) are atomic lattices, and classify their geometricity.