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

AbstractLet Γ denote a d-bounded distance-regular graph with diameter d⩾2. A regular strongly closed subgraph of Γ is said to be a subspace of Γ. Define the empty set ∅ to be the subspace with diameter -1 in Γ. For 0⩽i⩽i+s⩽d-1, let L(i,i+s) denote the set of all subspaces in Γ with diameters i,i+1,…,i+s including Γ and ∅. 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 LO(i,i+s) (resp. LR(i,i+s)). In the present paper we show that both LO(i,i+s) and LR(i,i+s) are atomic lattices, and classify their geometricity