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

Abstract

Let $$(L,\wedge ,\vee )$$ be a lattice with a least element 0. The annihilating-ideal graph of L, denoted by $${\mathbb {AG}}(L)$$ , 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 $$I\wedge J=\{0\}$$ . In this paper, we completely determine all finite lattices L with projective annihilating-ideal graphs $${\mathbb {AG}}(L)$$ .