Zero-divisor graphs of partially ordered sets

✍ Scribed by Zhanjun Xue; Sanyang Liu

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 448 KB
• Volume: 23
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2009.12.002

✦ Synopsis

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