The Embedding of Annihilating-Ideal Graphs Associated to Lattices in the Projective Plane

Let (L,perpendicular to,) be a lattice with a least element 0. The annihilating-ideal graph of L, denoted by AG(L), is a graph whose vertex set is the set of all non-trivial ideals of L and, for every two distinct vertices I and J, I is adjacent to J if and only if I perpendicular to J={0}. In this paper, we completely determine all finite lattices L with projective annihilating-ideal graphs AG(L).