Intersecting antichains and shadows in linear lattices

We establish a homomorphism of finite linear lattices onto the Boolean lattices via a group acting on linear lattices. By using this homomorphism we prove the intersecting antichains in finite linear lattices satisfy an LYM-type inequality, as conjectured by Erdős, Faigle and Kern, and we state a Kruskal–Katona type theorem for the linear lattices.