The Zero-Divisor Graph of a Lattice

Abstract

For a finite bounded lattice £, we associate a zero-divisor graph G(£) which is a natural generalization of the concept of zero-divisor graph for a Boolean algebra. Also, we study the interplay of lattice-theoretic properties of £ with graph-theoretic properties of G(£). (2010). Primary 05C75, 06E99; Secondary 13M99. Mathematics Subject Classification

Zero Divisor Graph of a Lattice and Its Unique Ideal

“…a ≤ y as given in Definition 3.6 of [2]. e base of k ∈ L is denoted by B(k) and is defined to be the set of all atoms a of L with a ≤ k…”

“…en, Γ(B) � K 1,1 .Proof. By Lemma 5.6 of[2], Γ(B) � K r,s ,. where r is the number of elements in a set ([a]↑\[b]↑) and s is the number of elements in a set ([b]↑\[a]↑).…”

“…In [1], the authors have introduced the notion of coloring in graphs derived from lattices. In [2], the authors associated to any finite lattice L a simple graph G(L) whose vertex set is Z(L) * , and two vertices x and y are adjacent if and only if x ∧ y � 0.…”

“…A nonempty subset I of L is called an ideal if a, b ∈ I implies that a ∨ b ∈ I and for l ≤ a implies l ∈ I. A proper ideal I of L is said to be a prime ideal if a, b ∈ L and a ∧ b ∈ I implies a ∈ I or b ∈ I. e undefined terms and notations are as in [2,3]. Let G � (V, E) be a graph.…”

The annihilating-ideal graph of a lattice

“…Several classes of graphs associated with algebraic structures have actively been investigated. For example, Cayley graphs have been studied in [10][11][12], zero-divisor graphs have been studied in [2][3][4]9], and cozero-divisor graphs have been introduced in [1].…”