Lattices generated by strongly closed subgraphs in dd-bounded distance-regular graphs

Let ΓΓ be a dd-bounded distance-regular graph with d≥3d≥3. Suppose that P(x)P(x) is a set of strongly closed subgraphs containing xx and that P(x,i)P(x,i) is a subset of P(x)P(x) consisting of the elements of P(x)P(x) with diameter ii. Let L(x,i)L(x,i) be the set generated by the intersection of the elements in P(x,i)P(x,i). On ordering L(x,i)L(x,i) by inclusion or reverse inclusion, L(x,i)L(x,i) is denoted by LO(x,i)LO(x,i) or LR(x,i)LR(x,i). We prove that LO(x,i)LO(x,i) and LR(x,i)LR(x,i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of P(x)P(x) on ordering P(x)P(x) by inclusion or reverse inclusion.