Planar zero divisor graphs of partially ordered sets

We associate a graph G∗(P) to a partially ordered set (poset, briefly) with the least element 0, as an undirected graph with vertex set P∗=P∖{0} and, for two distinct vertices x and y, x is adjacent to y in G∗(P) if and only if {x,y}ℓ={0}, where, for a subset S of P, Sℓ is the set of all elements x∈P with x≦s for all s∈S. We study some basic properties of G∗(P). Also, we completely investigate the planarity of G∗(P).