Strongly annihilator ideal graph of a lattice

Abstract

For a lattice [Formula: see text], we associate a graph called the strongly annihilator ideal graph of [Formula: see text], which is denoted by [Formula: see text]. It is a graph with vertex set consists of all ideals of [Formula: see text] having nontrivial annihilators, and any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if the annihilator of [Formula: see text] contains a nonzero element of [Formula: see text] and the annihilator of [Formula: see text] contains a nonzero element of [Formula: see text]. We show that [Formula: see text] is connected with the diameter at most two and its girth is 3, 4, or infinity. We characterize all lattices whose [Formula: see text] is planar. Among other results, it is proved that [Formula: see text] is perfect.