Abstract
Let L be a lattice with the least element 0. Let AL be the finite set of atoms with AL>1 and ΓL be the zero divisor graph of a lattice L. In this paper, we introduce the smallest finite, distributive, and uniquely complemented ideal B of a lattice L having the same number of atoms as that of L and study the properties of ΓL and ΓB.
Highlights
Let L be a lattice with the least element 0
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
In [3], the authors associated a simple, undirected graph Γ(L) with a lattice L with the vertex set Z(L)∗ Z(L) − {0}; the set of nonzero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y 0. ey studied the structure of Γ(L) and some basic properties of the zero divisor graph of a lattice. e zero divisor graph of various algebraic structures has been studied by several authors [4,5,6]
Summary
Let L be a lattice with the least element 0. We associate a simple graph Γ(L) to L with the vertex set Z(L)∗ Z(L) − {0}; the set of nonzero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y 0. 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. In [3], the authors associated a simple, undirected graph Γ(L) with a lattice L with the vertex set Z(L)∗ Z(L) − {0}; the set of nonzero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y 0. A complete bipartite graph is a graph whose vertices can be partitioned into two subsets P and Q such that no edge has both endpoints in the same subset, and every possible edge that
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