A graph associated to a lattice

Abstract

In this paper, we associate a simple graph to a lattice L, in which the vertex set is being the set of all elements of L, and two distinct vertices x and y are adjacent if x ∨ y ∈ S, when S is a multiplicatively closed subset of L. We denote this graph by S (L). We study some properties of S (L). Moreover, we investigate the planarity of S (L), whenever S is a saturated multiplicatively closed subset of L.

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

“…Researchers try to define a graph illustration for some kind of mathematical aspects. For example see [3] in the lattice literature, [13] in the measure literature and [16] in the real-valued continuous functions literature. The study of translating graph properties to algebraic properties is an interesting subject for mathematicians.…”

Graphsof BCI/BCK -algebras

“…Example 2.4 [2] Figure 1 shows that the 3-cube and complete graph K 4 are planar. Definition 2.5 [10] Let G be a plane graph.…”

“…He introduced the associated graph G to a ring R so that the vertex set of G contains elements of R and two vertices x and y of R are connected to each other if and only if xy = 0 . Many mathematicians, such as Anderson and Naseer [5], Anderson and Livingston [6], Maimani and Pournaki [4,18], Yassemi and Khashyarmanesh [2,3,21], and Torkzadeh and Ahadpanah [22], investigated the interplay between properties of the algebraic structure and graph theoretic properties. Motivated by these works, in this paper we study the associated graphs of BCI/BCK -algebras as two classes of abstract algebras, as introduced by Imai and Iseki [13,14] in 1996.…”

A graph associated to a commutative semiring

“…Over the last few years, the study of algebraic structures by graphs has been done and several interesting results have been obtained (see [1,2,4,5,10,11,[13][14][15][16][17]). For instance, the total graph of a commutative ring R is a simple graph whose vertex set is R, and two distinct vertices a and b are adjacent if a + b is a zero divisor of R (the set of all zero-divisor elements of R is denoted by Z(R))(see [3,18]).…”

“…(2) Let H be saturated. Then R \ H = i∈Λ M i , for some prime ideals M i of R, by (1). Let a ∈ H and r ∈ R. If r + a / ∈ H, then r + a ∈ M i , for some i ∈ Λ.…”