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`Journal of MathematicsVolume 2019, Article ID 9825495, 5 pageshttps://doi.org/10.1155/2019/9825495`
Research Article

## Zero Divisor Graph of a Lattice and Its Unique Ideal

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

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

#### 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 , the authors have introduced the notion of coloring in graphs derived from lattices. In , 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 , 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 (, 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 . The base of is denoted by and is defined to be the set of all atoms a of L with and .

Lemma 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. (, 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 , ,. 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 , , 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 . Following example shows that for nonatomic lattice L with the least element 0 and at least two atoms, .

Example 4. By Example 2 of , 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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