Abstract
Let be a multiplicative lattice and be a lattice module over . In this paper, we assign a graph to called residual division graph RG(M) in which the element is a vertex if there exists such that and two vertices are adjacent if (where ). It is proved that such a graph with the greatest element which does not belong to the vertex set is nonempty if and only if is a prime lattice module. Also, we provide conditions such that is isomorphic to a subgraph of Zariski topology graph with respect to .
1. Introduction
In our everyday life, we found that numerous issues are dealt with the assistance of graphs. Extraordinarily, the idea of the coloring of graphs assumes a significant function in computer sciences. To examine the coloring of rings, I. Beck first presented the zero-divisor graphs of a commutative ring with unity (see [1]). This examination of the coloring of a commutative ring was further studied by Anderson and Naseer (see [2]).
The ring structure is firmly associated with ideals more than elements and so it has the right to present a graph with vertices as ideals instead of elements. In this direction, M. Behboodi et al. studied the annihilating-ideal graph with vertex set contain all those ideals of ring whose annihilators are nonzero(see [3, 4]). Thereafter, Ansari-Toroghy et al. expanded this work for -module , where is a commutative ring. They studied the algebraic as well as topological properties of with the help of annihilating-submodule graph and the Zariski topology graph (see [5]). Recently, the idea of the zero divisor has likewise been applied in Boolean algebra, poset, and lattices (see [6, 7, 78]).
A complete lattice is a multiplicative if there is a defined binary operation called multiplication, denoted by “·”, which is commutative and associative such that the greatest element works as the multiplicative identity and for an arbitrary index set , where and is the least element of . It is interesting to note that the purpose behind the development of multiplicative lattice is to generalize lattices of ring ideals (see [9, 10]). Also, it is observed that the annihilating-ideal graph of a commutative ring with unity has close ties with a multiplicative lattice of ideals of . As far as the study of Johnson [11] is concerned, a lattice module is just an extension of a multiplicative lattice. It becomes worthy to study the graph over a commutative ring with unity with the help of a lattice module over .
Definition 1 (see [12]). A lattice module over the multiplicative lattice is a complete lattice with least element and greatest element if a multiplication between elements of and , represented by , where and , which satisfies the following properties:(1)(2)(3)(4), for all and for all Note that, for and , we define and . Here, the operation is called residual division (see [12]). Also, note that with ; the interval which is denoted by is a lattice module over a multiplicative lattice with the multiplication , where (see [12]).
Furthermore, for more definitions and concept of lattice modules and multiplicative lattice, see [9–19].
The semicomplement graph of lattice module introduced and investigated by Phadatare et al. (see [20]). In the recent paper [5], Ansari-Toroghy and Habibi have highlighted the closed sets in Zariski topology on prime spectrum of -module and defined new graph called Zariski topology graph, where which is nonempty. They studied the relationship between and annihilating-submodule graph (see [5]). Thereafter, this graph generalized to lattice modules over a -lattice and the Zariski topology graph was studied (see [21]).
Throughout the paper, denotes a lattice module over a multiplicative lattice , and for , we define .
The aim of this paper is to generalize the annihilating-submodule graph of a module to the lattice module over a multiplicative lattice and introduce the residual division graph whose vertex set is and in ; two vertices are adjacent if and only if . Apart from this, we will investigate interrelationship between and , where is the meet of all elements in .
Throughout the paper, denotes a lattice module over a multiplicative lattice .
2. Some Graph Theoretic Notions
We consider only undirected graphs. Thus, we adopt the notation , where is the set of vertices and is the set of edges of . A graph is an empty if . The number of edges incident on a vertex is called a degree of vertex and it is denoted by . A graph is said to be -regular if the degree of each vertex in is . Distance between the vertices and is the length of the shortest path between them, which is denoted by . Consider if there is no path between and . is the diameter of a graph . Length of shortest cycle in is called the girth of , denoted by . A clique of graph is its maximal complete subgraph, and the minimum number of cliques required to cover all the vertices of graph is called the partition number, denoted by . In a graph , a subset is supposed to be independent if no two vertices in are adjacent. The size of maximum independent set in a graph called as independence number, denoted by . For a vertex , denotes the set of all neighbors of in . A graph is said to be perfect if , for every induced subgraph of . Strongly perfect and very strongly perfect graphs are the classes of the perfect graph.
For further information, the reader may refer [22, 23].
3. Residual Division Graph
Definition 2. The residual division graph of is a graph with vertices , where distinct vertices and are adjacent if and only if .
Example 1. Figure 1 represents the residual division graph of with the vertex set , where represents lattice module over (see Figure 2 and 3).
We essentially need the following two Lemmas throughout this article.
Lemma 1 (see [12]). For and , the following holds:(1)If , then (2)(3)(4)(5)(6)If , then
Lemma 2 (see [12]). For , .
The following lemma gives a condition under which a nonzero proper element of lattice module is a vertex of .
Lemma 3. Let be a proper element of . Then, if or .
Proof. Suppose that is a proper element of and . Then, . Now, let . Therefore, . This implies that ; consequently, .
Callialp and Tekir [14] introduced the notion of multiplication lattice modules.
Definition 3 (see [14]). A lattice module is said to be multiplication lattice module if for each there exists an element such that .
Lemma 4 (see [14]). A lattice module is a multiplication lattice module if and only if , for all .
Lemma 5. Let be a proper element of multiplication lattice module . Then, if and only if .
Proof. Suppose that is a vertex of . Then, by definition, there exists such that . Therefore, . Since is multiplication, by Lemma 4, ; therefore, . This implies . However, ; therefore, . Converse part follows from Lemma 3.
Co-multiplication lattice module is introduced and characterized by F. Callialp et al. (see [16]).
Definition 4 (see [16]). Lattice module is called a co-multiplication lattice module if for each , there exists an element such that .
The following characterization plays an important role in the study of residual division graph .
Lemma 6 (see [16]). Lattice module is a co-multiplication if and only if for every element .
The following theorem is the immediate consequence of Lemma 6.
Theorem 1. Every proper element of a co-multiplication lattice module is a vertex of .
Proof. Let is a proper element of co-multiplication lattice module . By Lemma 3, to prove , we have to show that or . Suppose that . Then, by Lemma 1(6), . Since is a co-multiplication lattice module over a -lattice , by Lemma 6, which is contradiction to ; consequently, .
In the above three results, we studied various conditions on lattice module under which proper elements becomes a vertex. However, then the natural question arises: can the greatest element be a vertex in residual division graph
The following lemma answers the above question.
Lemma 7. Greatest element of is a vertex if and only if there exists a proper element such that .
Proof. Suppose that greatest element of a lattice module is a vertex. Then, there exists a such that . Therefore, . However, ; therefore, , and hence, . Conversely, suppose that , where is a proper element of with . Then, because of . Since , we have . By Lemma 1(5), ; therefore, ; hence, the greatest element is a vertex.
Theorem 2. Let the greatest element of is not a vertex of . Then, is a vertex if and only if .
Proof. Suppose that is in . Then, there exists such that . Therefore, and so . Since is not a vertex, by Lemma 7, , and hence, . Converse follows from Lemma 3.
In [13], Al-Khouja studied the relationship between the maximal (prime) elements of lattice module and the maximal (prime) elements of multiplicative lattice . “If is a prime element of a lattice module over a multiplicative lattice , then is a prime element of multiplicative lattice (see [13]).”
According to Callialp et al. (see [15]), a lattice module over a multiplicative lattice is prime if the least element is prime element of .
The following characterization is done by Callialp et al. (see [15]).
Lemma 8 (see [15]). Least element of is a prime if and only if , for all .
Lemma 8 helps us to characterize the prime lattice module.
Theorem 3. Let the greatest element of is not a vertex of . Then, if and only if is a prime lattice module.
Proof. Suppose that and is not a prime lattice module over . Then, least element is not a prime element of . Therefore, by Lemma 8, there exists proper element such that , and hence, by Lemma 3, is in , contradiction to ; consequently, is prime. Conversely, is a prime, and there exists such that . Then, by definition, there exists such that . This implies that , so . Since is a prime element of , prime element of . Therefore, or . This follows by Lemma 7 that is a vertex, a contradiction. Consequently, .
Theorem 4. For given , is connected and .
Proof. Suppose that such that . If , by definition, we have a path of length one. Now, suppose that . Case (1): if and , then and . This implies that is adjacent to and is adjacent to , i.e., is a path of length equal to 2. Case (2): if and , since , there exists such that . If , then we have a path with . Suppose that . Then, , and therefore, is a path of length 2 because . Similarly, if and , then we have a path with length 2. Case (3): if , , and , by definition, there exist such that . If , then is a path of length 2. Now, suppose that and . Since , we have a path of length 3, i.e., . Above cases implies that ; consequently, .
Corollary 1. If contain a cycle, then .
In [21], Phadatare et al. introduced the quasi-prime element of .
Definition 5 (see [21]). A proper element is said to be quasi-prime if is a quasi-prime element of .
The following lemma follows from Theorem 3.
Lemma 9. Let the greatest element of is not a vertex of . If, then the least element is a quasi-prime element of .
is a collection of all quasi-prime elements of and for , set (see [21]).
We basically need the following lemma.
Lemma 10 (see [21]). Let be a lattice module and . Then, if and only if .
Phadatare et al. [21] employed closed set to introduce the Zariski topology graph with respect to .
Definition 6. For , we define an undirected graph associated with , called Zasiski topology graph with respect to with vertex set and distinct vertices and are adjacent if and only if .
Remark 1. By Proposition 3.1 of [21] and Lemma 2, for , .
Note that, for , an interval denoted by is a lattice module over a multiplicative lattice with the multiplication , where .
Theorem 5. Let be a lattice module, and is not a vertex of . Then, is isomorphic to subgraph of .
Proof. Suppose that . By definition, there exists such that is adjacent to . Therefore, . Note that . Therefore, we have . This impies that ; therefore, by Lemma 1(1), , and hence, by Lemma 10, . Since , we have . If , then ; therefore, by Lemma 7, is a vertex of which is a contradiction; consequently, . In the same line, we have . Thus, such that is adjacent to .
Corollary 2 (see [21]). For , is nonempty if and only and.
The following is the characterization of nonempty residual division graph .
Theorem 6. If is not a vertex of , then if and only if .
Proof. Suppose that is not a vertex of and . Then, by Lemma 9, is a quasi-prime element of ; therefore, by Corollary 2, . Converse follows from Theorem 6.
Theorem 7. The greatest element is a vertex in if there exists such that and .
Proof. Suppose there exists such that and . To prove that , it suffices to prove that there exists such that . Note that . By definition, Since , we have, for all , ; therefore, , and hence, by Lemma 1(2), . This implies that , i.e., ; therefore, ; consequently, is a vertex in .
For , represents -radical of , and it is defined as . If , then is said to be -radical element of .
“If the natural map defined by surjective, then (see [24]).”
Theorem 8. If the natural map is surjective and are adjacent vertices in . Then, and are adjacent in .
Proof. To show that and are adjacent in , we have to prove that and with . Suppose that are adjacent vertices in . Then, we have with . However, ; therefore, , and hence, . Thus, . Also, note that . Since is surjective and , we have ; therefore, . However, ; therefore, . Now, it remains to prove that and . If , then . Therefore, , and hence, . By Lemma 10, , a contradiction. Consequently, . Similarly, we have .
4. Conclusion
In this paper, we introduced the residual division graph of the lattice module and characterized the co-multiplication lattice module and prime lattice module. Also, we characterized vertex set of residual division graph of the multiplication lattice module. We found that the residual division graph of an interval lattice module is isomorphic with the Zariski topology graph.
Data Availability
The data from previous studies were used to support this study. They are cited at relevant places within the article as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.