Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 409 https://internationalpubls.com Prime Spectrum Graph of C-Lattices Lakpa Sherpa1, Vilas Kharat2, Manish Agalave3, Ganesh Gandal4, Narayan Phadatare5 1,2Department of Mathematics, Savitribai Phule Pune University, Pune-411 007 (INDIA) 3Department of Mathematics, Fergusson College(Autonomus), Pune-411 004 (INDIA) 4Smt Kashibai Navale College of Engineering, Pune (INDIA) 5Bharati Vidyapeeth Deemed to be University College of Engineering, Pune-411 043 (INDIA) csherpaap@gmail.com,laddoo1@yahoo.com,manishagalave@gmail.com,ganeshgandal2001@gmail.com nmphadatare@bvucoep. Article History: Received: 18-02-2024 Revised: 20-04-2024 Accepted: 12-05-2024 Abstract: Let £ be a C-lattice. Let σ(£) be the set of all prime elements of £ and M(£) be the collection of all maximal elements of £. For S ⊆M(£), we introduce the new graph called S-join graph on σ(£), denoted by ΓS(σ(£)). We have studied properties like connectivity, diameter and domination number of the graph σ(£) . In this paper, we established that the topological space σ(£) is connected if and only if the graph ΓS(σ(£)) is connected. Keywords: Prime element; Maximal element; S-join graph. 1. INTRODUCTION The study of commutative rings indeed enriched by using graph theory techniques. In 1988, I. Beck proved that how graph theory can be applied to the study of commutative rings. According to I. Beck, the zero-divisor graph of a commutative ring is a graph where the vertices represent the elements of the ring, and two vertices are connected by an edge if their product is zero (see [5]). Further, by employing these graph-theoretic approach, many researchers visually represented and analyzed various aspects of commutative rings, providing additional insights into their structure, properties, and relationships (see [1]-[4], [9]-[10], [12]). The ideals of ring play a fundamental role in the study of ring structure. Therefore M. Behboodi et. al. introduced and studied annihilating-ideal graph whose vertices are annihilating ideals of a commutative ring with unity (see [6]-[7]). The set of ideals of a ring is naturally endowed with a lattice structure. It is very much interesting that the set of ideals of a ring, denoted as Id(R), forms a multiplicative lattice. Definition 1.1. A multiplicative lattice is denoted as (£,0,1,∗), where £ is a complete lattice with least element 0, greatest element 1 and ∗ is a binary operation defined on £ that satisfies the following properties for all a,b,c ∈ £: 1. a ∗ b ≤ a ∧ b. 2. a ∗ b = b ∗ a. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 410 https://internationalpubls.com 3. (a ∗ b) ∗ c = a ∗ (b ∗ c). 4. a ∗ (∨α∈Ibα) = ∨α∈I(a ∗ bα), where bα ∈ £ and I is an indexing set. 5. a ∗ 1 = a. Henceforth, we write a ∗ b = ab for the sake of convenience only. A member a ∈ £ is called compact if a ≤ ∨β∈I aβ implies a ≤ ⋁𝑖=0 𝑛 𝑎β𝑖 . Let £c the set of all compact elements of £. A multiplicative lattice £ is called compactly generated if each a ∈ £ is of the form ∨bi for bi ∈ £c. By C-lattice £, we mean a multiplicative lattice (£,0,1,•) which is generated under join by a multiplicatively closed set C of compact elements and the greatest element 1 is compact as well as multiplicative identity. An element p ∈ £ is said to be proper if p <1. In a multiplicative lattice £, a proper element m is maximal, if m is not properly contained within any other element of £ under the partial order relation ≤. We denote M(£), the collection of all maximal elements of £. In a multiplicative lattice £, if greatest element 1 is compact then each a <1 lies below some m ∈ M(£). If M(£) = {m}, then £ is called as local. For a,b ∈ £, a proper element p ∈ M is said to be prime, if ab ≤ p, then a ≤ p or b ≤ p. Let σ(£) the collection of all prime elements of £. As each maximal element is prime, we have M(£) ⊆ σ(£). A multiplicative lattice £ is said to be domain if 0 ∈ σ(£). In this paper, we used a non-empty subset S of M(£) and defined a new simple, undirected graph called the S-join graph ΓS(σ(£)) with the vertex set σ(£), where σ(£) is the collection of all prime elements of £ and two distinct vertices a and b are adjacent i.e., a ∼ b if and only if a ∨ b ≤ m for some m ∈ S. Here, we study some basic properties like connectivity, girth and clique number of the graph ΓS(σ(£)). Throughout this paper, multiplicative lattice £ assumed to be C-lattice £. By σ(£) and M(£), we mean the collection of all prime elements, and the collection of all maximal elements of £, respectively. 2. GRAPH THEORETIC DEFINITIONS Let the undirected graph G = (V,E), where V = V (G) is the set of vertices of G and E = E(G) is the set of edges of G. A graph with empty vertex set is called an empty graph. Let b ∈ V , the number of edges incident on b is called degree of a vertex b and it is denoted by deg(b). In a graph G, d(a,c) represents the length of shortest path between a and c. Note that, d(a,c) = ∞, if there is no path between a and c. The diameter of a graph G is defined as diam(G) = sup{d(a,c)|a,c ∈ V (G)}. The length of shortest cycle in G is called the girth of G, denoted by gr(G). A clique of graph is its maximal complete subgraph. For a graph G, a subset S ⊆ V (G) is supposed to be independent, if no two vertices in S are adjacent. The independence number α(G) is the maximum size of an independent set in G. Let ∅ ≠ S ⊆ V. If each vertex in V−S is adjacent to some vertex in S, then S is called a dominating set. Number of vertices in smallest dominating set is called domination number and it is denoted by γ(G). For more information on graph theory, the reader may refer ([11], [14]). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 411 https://internationalpubls.com 3. S-JOIN GRAPH ΓS((σ(£)) We start this section with the following definition. Definition 3.1. Let M(£) be the set of all maximal elements of C-lattice £ and ∅ ≠ S ⊆ M(£). The S-join graph ΓS(σ(£)) is simple, undirected graph with vertex set σ(£) and two distinct vertices a and b are adjacent if and only if a ∨ b ≤ m for some m ∈ S. Example 3.2. The lattice in Figure (1) is a multiplicative lattice £ and Figure (2) represents the S-join graph ΓS(σ(£)) with the vertex set σ(L) = {a,c,d} and S = {c,d}. Figure (1) Multiplicative Lattice £ a d c Figure (2) ΓS(σ(£)) Definition 3.3. A non-empty subset S of M(£) is said to be small with respect to ∧ in short ∧- min set, if for any a,b ∈ S, there is no 0 ≠ p ∈ σ(£) such that p ≤ a ∧ b. Proposition 3.4. Let £ be a C-lattice and S be a ∧-min set. If the S-join graph ΓS(σ(£)) is connected, then £ is a local or a domain. Proof. Suppose that zero element of £ is not a prime and m1,m2 ∈ M(£). It is given that the S- join graph ΓS(σ(£)) is connected, we have a path m1 ∼ p1 ∼ ••• ∼ pn ∼ m2 between m1 and m2. Clearly, all pi’s are non-zero members of σ(£). Since m1 is adjacent to p1 and p1 is adjacent to p2, we have p1 ≤ m1 and there exists m ∈ S with p1∨p2 ≤ m. This implies that, p1 ≤ m1 ∧ m. Since S is ∧-min set and p1 ≠ 0 implies that m1 = m. By the similar arguments, we have m1 = m2. Hence, £ is a local. Proposition 3.5. If £ is a C-lattice which is either a local or a domain S is non-empty subset of M(£), then the S-join graph ΓS(σ(£)) is connected. Proof. Suppose that £ is a C-lattice which is a domain, then 0 ∈ σ(£). Therefore for any prime elements p and q other than 0, we have, a path p ∼ 0 ∼ q. Hence, ΓS(σ(£)) is connected. Now, if £ is a C-lattice which is a local with M(£) = {m}. Then for any p1,p2 ∈ σ(£), we have p1 ∨ p2 ≤ m. Therefore, p1 ∼ p2. Consequently, ΓS(σ(£)) is connected. 0 a b c d 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 412 https://internationalpubls.com From the Proposition 3.5, we have following corollary. Corollary 3.6. Let £ be a C-lattice. If £ is a local or a domain, then diam(ΓS(σ(£))) ≤2. In the following Theorem, we have obtained a characterization of a local C-lattice £. Theorem 3.7. Let £ be a C-lattice and let ΓS(σ(£)) be the S-join graph of £. Then the graph ΓS(σ(£)) is complete if and only if £ is a local. Proof. Suppose that, ΓS(σ(£)) is a complete graph of a C-lattice £ with m1,m2 ∈ M(£). Since the graph ΓS(σ(£)) is complete and every maximal element of £ is prime, we have m1 ∼ m2, implies that m1∨m2 ≤ m for some m ∈ M(£) such that m ≠ 1, a contradiction to the fact that m1 ∨ m2 = 1 so that m1 = m2. Hence, £ is a local C-lattice. Conversely, suppose that £ is a local C-lattice with the maximal element m. Then for any p1,p2 ∈ σ(£), we have p1 ∨ p2 ≤ m and hence p1 and p2 are adjacent. Consequently, the S-join graph ΓS(σ(£)) is complete. Proposition 3.8. If a C-lattice £ is such that M(£) = σ(£) − {0}, with unique atom, then ΓS(σ(£)) is a star graph. Proof. Suppose that a C-lattice £ is a domain. so that, 0 ∈ σ(£). Also, any two non-zero elements a,b ∈ £ are co-maximal. Therefore, for any 0 ≠ a ∈ £, we have a path 0 ∼ a. Also, there does not exist edge between any two non-zero elements of £. Hence, ΓS(σ(£)) is a star graph. The following definition of dimension of a C-lattice £ is used in results that follows. Definition 3.9. Let £ be a C-lattice. The dimension of £, denoted as dim(£), is the supremum of the lengths of chains of members of σ(£). Theorem 3.10. Let £ be a C-lattice which is not a local and let S be a ∧-min set of £. If a S- join graph ΓS(σ(£)) is a star graph, then £ is a domain with dim(£) ≤ 1. Proof. Suppose that ΓS(σ(£)) is a star graph of £. Hence, there is a vertex p in the graph ΓS(σ(£)) such that p ∼ q, for all q ∈ σ(£). Therefore, for m1,m2 ∈ M(£), we have p ∼ m1 and p ∼ m2. Since m1,m2 ∈ M(£), p ∨ m1 = m1 and p ∨ m2 = m2. This implies that p ≤ m1 ∧ m2. But it is given that S is a ∧-min set, therefore p = 0. Hence, 0 is a prime element of £, consequently, £ is a domain. Now, suppose that dim(£) ≥ 2, then there exists a chain of prime elements having length at least two, say, p1 ≤ p2 ≤ p3. It is clear that p1 = 0. Since ΓS(σ(£)) is a star graph, we have p1 ∼ p2, by definition there exists m ∈ M(£) such that p1 ∨ p2 ≤ m, i.e., p1 ∨ p2 = p2 ≤ m. If p2≠ m, then we have a cycle p1 ∼ p2 ∼ m ∼ ••• ∼ p1, a contradiction to the fact that the graph ΓS(σ(£)) is a star graph. If p2 = m, then we have p1 ≤ p2 = m ≤ p3 which not possible because m is maximal. Consequently, dim(£) ≤ 1. Proposition 3.11. Let £ be a C-lattice and ΓS(σ(£)) be a S-join graph of £. If ΓS(σ(£)) is a star graph, then S = M(£). Proof. Suppose that the S-join graph ΓS(σ(£)) of £ is a star graph. Then, we have a fixed vertex, say a, in the graph ΓS(σ(£)) such that a ∼ b for each vertex b in ΓS(σ(£)). Therefore, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 413 https://internationalpubls.com for any m ∈ M(£), we have a ∼ m. This implies that, a ∨ m ≤ m1, for some m1 ∈ S. As, m ∈ M(£), implies that m = m1. Hence, S = M(£). We need the following proposition to establish the some important theorem ahead. Proposition 3.12. [13] Let £ be a multiplicative lattice with greatest element 1 is compact. Then for any a ∈ £ such that a ≠ 1, there exists m ∈ M(£) such that a ≤ m. Theorem 3.13. Let £ be a C-lattice and ΓS(σ(£)) be a S-join graph of £. If dim(£) ≥ 2, then gr(ΓS(σ(£))) = 3. Proof. Suppose that a0 ≤ a1 be a chain of members of σ(£). Since £ is a C-lattice, we have greatest element 1 is compact. Therefore, by Proposition 3.12, there exists a maximal element m such that a1 ≤ m. Now, we have a0 ∼ a1 ∼ m ∼ a0, since a0 ∨ a1 ≤ m and a0 ∨ m = m. Theorem 3.14. Let £ be a C-lattice with dim(£) = 1. If a S-join graph ΓS(σ(£)) of £ contains a cycle, then £ is not a domain. Proof. Suppose on the contrary that 0 ∈ σ(£). Since dim(£) = 1 and graph ΓS(σ(£)) of £ contains a cycle, by Theorem 3.13 we have a cycle a0 ∼ a1 ∼ a2 ∼ ••• ∼ a0 in ΓS(σ(£)). As a0 ∼ a1, there exist m ∈ M(£), such that a0 ∨ a1 ≤ m. Here either a0 or a1 are non-zero member of σ(£). Suppose that a0 ̸= 0. Since 0 ∈ σ(£) and dim(£) = 1, we have a1 is maximal element. In fact, a1 = m. Similarly, we can prove that a2 is also maximal element. Hence a1 and a2 are co- maximal elements of £, which is contradiction to the fact that a1 ∼ a2. Consequently, 0 ∈ σ(£). Note, σmin(£) denotes the set of all minimal prime elements of £. Theorem 3.15. Let £ be a C-lattice with |σmin(£)| ≤ ∞. Then γ[ΓS(σ(£))] ≤ |σmin(£)|. Proof. Suppose on the contrary that κ = {p1,p2,•••,pn} be a dominating set in the S-join graph ΓS(σ(£)) of £. By definition, for any q ∈ σ(£) − κ there is pi ∈ κ such that q is adjacent to pi for some 1 ≤ i ≤ n. Let q1,q2,••• ,qr be the distinct members of σmin(£). For each pi ∈ κ there is at least one qj (1 ≤ j ≤ r) with qj ≤ pi. This implies that r ≤ n. Now we prove that τ = {q1,q2,••• ,qr} is also a dominating set. Suppose that p ∈ σ(£) such that p ∉ τ. If p ∉ κ. Since κ is a dominating set, we have pi ∈ κ such that p ∼ p1. By definition there exists m′ ∈ M(£) such that p ∨ pi ≤ m′. According to the construction of the set τ = {q1,q2,••• ,qr}, there are some qj ∈ τ with qj ≤ pi, therefore p ∨ qj ≤ m′ and hence p ∼ qj for some qj ∈ τ. Now, suppose p ∈ κ. By the definition of τ, there exists qi ∈ τ such that qi ≤ p. Since the greatest element 1 is compact, by Proposition 3.12 we have m ∈ M(£) such that p ≤ m, therefore qi ∨ p ≤ m. This implies that qi ∼ p, consequently, τ is a dominating set. In [8], F. Callialp et. al. established some results on the Zariski topology over σ(£). For a ∈ £, define ϑ(a) = {p ∈ σ(£)|a ≤ p}. F. Callialp et. al. introduced a topology on σ(£) with the collection of all closed set {ϑ(a)|a ∈ £} using the following Proposition 3.16 (see [8]). Proposition 3.16. [8] Let £ be a C-lattice and for a ∈ £, let ϑ(a) = {p ∈ σ(£)|a ≤ p}. Then the following axioms hold: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 414 https://internationalpubls.com 1. ϑ(0) = σ(£) and ϑ(1) = ∅. 2. ∩α∈△ϑ(aα) = ϑ(∨α∈△aα) for any index set △. 3. ϑ(p) ∪ ϑ(q) = ϑ(p ∧ q) = ϑ(pq) for p,q ∈ £. In the next Theorem 3.17, we studied the connected topological space σ(£). Theorem 3.17. Let £ be a C-lattice and |σmin(£)| ≤ ∞. Then σ(£) is connected if and only if the S-join graph ΓS(σ(£)) of £ is connected and diam(ΓS(σ(£))) ≤ 2|σmin(£)|. Proof. Suppose that σ(£) is not connected. If ΓS(σ(£)) is disconnected, then nothing to prove. Suppose that ΓS(σ(£)) is connected, then for some a,b ∈ £, σ(£) = ϑ(a) ∪ ϑ(b) and ϑ(a) ∩ ϑ(b) = ∅. Let p,q ∈ σ(£) such that p ∈ ϑ(a) and q ∈ ϑ(b). Since ΓS(σ(£)) is the connected graph, we have a path p ∼ p1 ∼ p2 ∼ p3 ∼ ••• ∼ pn ∼ q between p and q. As p ∼ p1, by definition there exists m1 ∈ M(£) such that p ∨ p1 ≤ m1. This implies that p1 ∈ ϑ(a) since p,m1 ∈ ϑ(a). Using similar arguments, we can prove that pn ∈ ϑ(a). Also, note that pn ∼ q. Therefore by definition, there exists mn ∈ M(£) such that pn ∨ q ≤ mn. This implies that mn ∈ ϑ(a) because pn ≤ mn. Also, note that q ≤ mn, therefore we have mn ∈ ϑ(b), a contradiction to the fact that ϑ(a) ∩ ϑ(b) = ∅. Consequently, σ(£) is connected. Conversely, suppose that the space σ(£) is connected and σmin(£) = {a1,a2,••• ,ar}. Case I) Suppose r = 2. For p,q ∈ σ(£), there exist a1,a2 ∈ σmin(£) such that a1 ≤ p and a2 ≤ q. Since σ(£) is connected, we have a3 ∈ ϑ(a1)∩ϑ(a2). Also, since the greatest element 1 is compact, by Proposition 3.12 we have a path p ∼ a1 ∼ a3 ∼ a2 ∼ q of length 4 = 2|σmin(£)| between p and p. Case II) Suppose that r >2. Let p, q ∈ σ(£) such that for 1 ≤ i ≤ l1, p ∈ ϑ(ai), for l1 ≤ i ≤ l2, q ∈ ϑ(ai) and for l2 ≤ i ≤ r, p, q ∉ ϑ(ai). But σ(£) is connected, therefore there exist 1 ≤ i1 ≤ ••• ≤ ik ≤ r with ϑ(𝑎𝑖𝑙 ) ∩ ϑ(𝑎𝑖𝑙+1 ) ≠ ∅ for 1 ≤ l ≤ k and ϑ(∨𝑖=1 𝑙1 𝑎𝑖) ∩ ϑ(𝑎𝑖1 ) ≠ ∅, ϑ(∨𝑖=𝑙1+1 𝑙2 𝑎𝑖) ∩ ϑ(𝑎𝑖𝑘 ) ≠ ∅. This implies that there is a path between p and q with length at most 2|σmin(£)|. Consequently, the graph ΓS(σ(£)) of £ connected and diam(ΓS(σ(£))) ≤ 2|σmin(£)|. REFERENCES [1] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274(2)(2004),847-855. [2] D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings,and Boolean algebras, J.Pure Appl. Algebra, 180(2003), 221-241. [3] F.F. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. 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