Journal of Mathematics
Volume 2019 (2019), Article ID 9825495, 5 pages
https://doi.org/10.1155/2019/9825495

Research Article

Zero Divisor Graph of a Lattice and Its Unique Ideal

Rajendra Deore¹ and Pramod Tayade²

¹Department of Mathematics, Kavayitri Bahinabai Chaudhary North Maharashtra University, Jalgaon, Jalgaon-425001, India
²Department of Mathematics, University of Mumbai, Mumbai-400098, India

Correspondence should be addressed to Pramod Tayade; tayade.pramod@yahoo.in

Received 21 July 2019; Accepted 16 September 2019; Published 1 December 2019

Academic Editor: Feng Feng

Copyright © 2019 Rajendra Deore and Pramod Tayade. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let L be a lattice with the least element 0. Let be the finite set of atoms with and 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 and .

1. Introduction

Let L be a lattice with the least element 0. An element is said to be a zero divisor if there exists a nonzero element such that . The set of zero divisors in L is denoted by . We associate a simple graph to L with the vertex set ; the set of nonzero zero divisors of L and distinct are adjacent if and only if .

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 whose vertex set is , and two vertices x and y are adjacent if and only if .

In [3], the authors associated a simple, undirected graph with a lattice L with the vertex set ; the set of nonzero zero divisors of L and distinct are adjacent if and only if . They studied the structure of and some basic properties of the zero divisor graph of a lattice. The zero divisor graph of various algebraic structures has been studied by several authors [4-6].

We now give here some preliminaries. be a lattice with the least element 0. An element is called as an atom if there is no such that . The set of all atoms is denoted by . In this paper, we consider lattices with at least two atoms. The lattice L is called atomic if for any , there exists an element such that . A nonempty subset I of L is called an ideal if implies that and for implies . A proper ideal I of L is said to be a prime ideal if and implies or .

The undefined terms and notations are as in [2, 3]. Let be a graph. A graph G is said to be connected if for each pair of distinct vertices x and y, there is a finite sequence of distinct vertices such that each pair of vertices is an edge, and such a sequence of vertices is called as a path. The number of edges in a path is the length of a path. For distinct vertices x and y of G, let be the length of the shortest path from x to y. if there is no path from x to y and . The diameter of a graph G is . The length of a smallest cycle in a graph G is called as girth, and it is denoted by and if G contains no cycle. The degree of a vertex in G is the number of edges incident on which is denoted by . The maximum degree vertex of a graph G, . A graph G is said to be bipartite if the vertex set of G is partitioned into two disjoint subsets P and Q such that no edge has both end points in any one of P or Q. 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 could connect vertices in different subsets is part of the graph. However, in a complete bipartite graph G if , , then it is denoted by . A star graph is a complete bipartite graph . A graph in which each pair of distinct vertices is joined by an edge is a complete graph, and it is denoted by . A subset of L is called a clique in if are adjacent for all . The clique number is the cardinality of the maximum possible complete subgraph of . If the size of a clique is not bounded, then we say . A (vertex) coloring of G is assigning of colors to the vertices so that adjacent vertices have different colors. The minimum number of colors required to color G is the chromatic number of G and is denoted by .

2. Diameter and Line Graph on Four Vertices

Lemma 1. Let L be a lattice with the least element 0 and , the set of atoms in L. If , then .

Proof. Let . Then, and .

Proposition 1. A connected graph on four vertices with vertex set and edges cannot be realised as .

Proof. On the contrary, suppose that is the required lattice, it must be closed under meet. In particular, . We have the following cases:

 Case 1: implies that  Case 2: implies that  Case 3: implies that  Case 4: implies that

In each of the above cases, we get a contradiction. Thus, we have is not in L. Hence, L is not a lattice.

Proposition 2 ([3], Proposition 1.4). If L is a lattice, then .

Theorem 1. Let L be a lattice with the least element 0. If contains a cycle, then .

Proof. Assume that contains the smallest cycle of length n. If , then we are done. Suppose that . Therefore, . We have the following cases for consideration:

 Case 1: if , then is a cycle of length three smaller than . Similarly, . Case 2: if , then is a cycle of length , smaller than . Case 3: if , then it is similar to Case 2.

From Cases 1, 2, and 3, we conclude that , and . However, is a cycle of length 3, smaller than , a contradiction to our assumption. Hence, .
Let be a finite bounded lattice and . Then, we define , as given in Definition 3.6 of [2]. The base of is denoted by and is defined to be the set of all atoms a of L with and .

Lemma 2 ([2], Lemma 5.7). Let L be a finite bounded lattice and such that . Then, .
Following example shows that the converse of the above lemma is not true.

Example 1. For lattice , and in .

3. Unique Ideal of Lattices

Lemma 3. Let L be a lattice with a nonempty subset . Then, is the smallest ideal of L containing S.
An Ideal of a lattice L is a sublattice but not conversely.

Theorem 2. Let L be a nondiamond lattice with the least element 0 and having a finite set of atoms. Then there exists a finite, distributive, and uniquely complemented ideal B of L such that .

Proof. Let be the set of atoms of L. Let , where is the largest element of B which is a locally maximal element of a lattice L and 0 is the least element. By Lemma 3, B is finite and is the smallest ideal of L containing all atoms of L.

Corollary 1. Let L be a lattice with the least element 0 and having a finite set of atoms. Then, B is a finite, distributive, and uniquely complemented sublattice of L.

Corollary 2. Let L be a lattice with the least element 0 and has a finite set of atoms. Then,

(a)(b)

Proof. (a) For . (b) For , . If , then .

Corollary 3. Let L be a lattice with least element 0, be the set of all atoms of L, and a unique ideal of L. Let and 1 be the largest elements of B and L, respectively. Then,

(a)If , then x has no complement in L(b)If , then b has a complement in B but has no complement in L

Corollary 4. ([2], Corollary 5.11). Let a be an element of L such that and a be comparable with all elements of L. Then, .

Theorem 3. Let L be a lattice with the least element 0 and having a finite set of atoms. If B is the unique ideal, then is the induced subgraph of .

Proof. Proof follows by Corollary 2 and Corollary 4.

Theorem 4. Let L be a lattice with the least element 0 and having a finite set of atoms and B be its unique ideal. Then, is the n-partite induced subgraph of .

Proof. Let . Then, is the set of atoms in a finite sublattice B. Therefore, . There is no adjacency between any two arbitrary elements in , for all . If for some , then there is no adjacency between x and both elements and . Thus, there exist disjoint subsets of , . This shows that is n-partite.

Theorem 5. Let L be a finite lattice with . and . Then,

(1) for all (2)(3)(4)(5)

Theorem 6. Let L be a finite lattice with and and . Then,

(1)(2)(3)

Example 2. Let be a lattice under the divisibility relation and 1 is the least and 0 is the largest element of L. Here, is the unique ideal, and is not since 3 is the complement of 2 and 4.

Example 3. Let for positive integer k and let . Then, L is a lattice under divisibility relation and 1 is the least and 0 is the largest element of L. Here, is the unique ideal and and …. are not since 3 is the complement of more than two elements.

Proposition 3. Let L be a lattice with the least element 0. Let with a unique ideal . Then, .

Proof. By Lemma 5.6 of [2], ,. where r is the number of elements in a set and s is the number of elements in a set . Now and , . Similarly, .

Proposition 4. Let L be a lattice with the least element 0 and let . Then, a unique sublattice is isomorphic to a Boolean ring A.

Proof. Let B be the unique ideal of L. Then, by Definition 4.1 and Proposition 4.2 of [2], , where B is a Boolean ring under the operations and ( is a complement of ).

Proposition 5. Let be the lattices with the least element . Then, is a lattice.

Proposition 6. Let be the lattices with the least element . Then, .

Proof. An ordered k-tuple is an atom if at most one component is an atom in and the rest of the components are corresponding zero’s of . Hence, the result.

Proposition 7. Let L be a lattice with the least element 0 and B be its unique ideal. Let be the set of atoms in L. Then, .

Proof. Suppose that is a clique. Then, by definition of a unique ideal B, for some , and , shows that C is not a clique.

Theorem 7. Let L be lattice with the element 0 and having a finite set of atoms. Let B be its unique ideal. Then, if and only if .

Proof.

 Case 1: clearly . Let be the set of atoms in L. We decompose as , for and . Define a map on by for . If are adjacent, then , so is a coloring on ; hence, by Proposition 7. Conversely, assume that is the set of atoms of B is a clique in B. Then, . Case 2: if B has an infinite number of atoms, then ; hence, . Suppose . Let be an infinite clique in . Since B is atomic, for each i, there exists an atom such that . Again are adjacent for , and it shows that . Hence, B has an infinite number of atoms.

Proposition 8. Let L be a lattice with the least element 0 and B be its unique ideal. Then, .

Proof. By Theorem 7, , applying Theorem 7, for atomic lattice with n number of atoms, , similar to Theorem 2.1 in [1]. Following example shows that for nonatomic lattice L with the least element 0 and at least two atoms, .

Example 4. By Example 2 of [1], we know that the set is a bounded distributive lattice having no atom. Let be a lattice of divisors of 10 under the divisibility relation. Then, is nonatomic lattice with two atoms. The unique ideal . Here, .

Lemma 4. Let L be a lattice with the least element 0 and B be its unique ideal. If B contains an infinite chain, then .

Proof. Let be an increasing chain in B. Set . If , then by distributivity in B, we have , which contradicts the hypothesis. Suppose that and . Therefore, and , a contradiction. This shows that are distinct. Again implies that . Hence, is an infinite clique in L.

4. Diamond Lattices

An element is called a coatom if . A lattice L is said to be a diamond lattice if every element is an atom and coatom. A diamond with n atoms is denoted by .

Theorem 8. Let be a diamond. Then, is a distributive and uniquely complemented ideal of . Moreover, .

Theorem 9. Let be a diamond lattice. Then,

(1)For is a distributive and complemented ideal of (2)For (3)For

Theorem 10. Let be a lattice. Then, is a distributive and complemented ideal of L, which can be generalized.

5. Some Combinatorial Results

Theorem 11. If L is a lattice which is nondiamond or nondecomposable into diamonds with the least element 0 and , then in .

Proof. Let . . Then, in a finite subgraph , is adjacent to vertices; hence, contribution of is . Again is adjacent to , and contribution of for this pair is . Continuing in this way, contribution of for is . Total contribution is .

Corollary 5. Let L be a lattice with the least element 0 and B be its unique ideal. Then, for the induced subgraph ,

(1)(2)

Proposition 9. Let L be a finite lattice with the least element 0 with and . Then, .

Proof. Here, ; hence, for .

Proposition 10. Let L be a lattice with the least element 0, set of atoms, and B be the unique sublattice of L. Then, .

Proof. Let . For , zero divisors, their contribution is . Also are zero divisors, and their contribution is . Continuing is this way, contribution of is a set of zero divisors and is .

Corollary 6. Let L be a finite lattice with the least element 0 and B be its unique ideal. Then, .

Proof. Proof follows by Corollary 2 and Proposition 10.

Data Availability

Data from previous studies were used to support this study. They are cited at relevant places within the text as references.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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