Zero-divisor graphs of partially ordered sets

Let (P,≤)(P,≤) be a partially ordered set (poset, briefly) with a least element 0 and S⊆PS⊆P. An element x∈Px∈P is a lower bound of SS if s≥xs≥x for all s∈Ss∈S. A simple graph G(P)G(P) is associated to each poset PP with 0. The vertices of the graph are labeled by the elements of PP, and two vertices xx, yy are connected by an edge in case 0 is the only lower bound of {x,y}{x,y} in PP. We show that if the chromatic number χ(G(P))χ(G(P)) and the clique number ω(G(P))ω(G(P)) are finite, then χ(G(P))=ω(G(P))=n+1χ(G(P))=ω(G(P))=n+1 in which nn is the number of minimal prime ideals of PP.