Abstract
Rough sets are a key tool to model uncertainty and vagueness using upper and lower approximations without predefined functions and additional suppositions. Rough graphs cannot be studied more effectively when the inexact and approximate relations among more than two objects are to be discussed. In this research paper, the notion of a rough set is applied to hypergraphs to introduce the novel concept of rough hypergraphs based on rough relations. The notions of isomorphism, conformality, linearity, duality, associativity, commutativity, distributivity, Helly property, and intersecting families are illustrated in rough hypergraphs. The formulae of 2-section, L2-section, covering, coloring, rank, and antirank are established for certain types of rough hypergraphs. The relations among certain types of products of rough hypergraphs are studied in detail.
1. Introduction
In graphical networks, usually pairwise relations are discussed missing some information that more than two objects may satisfy common characteristics. Hypergraphs introduced by Berge [1] as a generalization of graphs tackle the difficulty to study relations and common characteristics of any set of objects. A lot of work has been done on hypergraphs due to their applications in various domains of biological and computer sciences including properties and algorithms of the Cartesian product of hypergraphs [2], the direct product of hypergraphs [3], and hamiltonicity of certain products of hypergraphs [4]. Hammack et al. [5] studied distance measures, isometries, factorization, chromaticity, and various other properties of graph products which is a strong framework to generalize all the results for hypergraphs.
Hypergraphs are the key tool to study real-world problems in a more generalized and efficient way as compared to graphs and their extensions but are unable to study uncertainty and vagueness occurring in data and incomplete information. Kaufmann [6] extended the concept of hypergraphs to fuzzy hypergraphs by applying fuzzy sets [7] on hypergraphs. Lee-Kwang and Lee [8] proved that there are some flaws in Kaufmann’s definition of a fuzzy hypergraph and redefined that concept. Radhamani and Radhika [9] added fuzzy relations in fuzzy hypergraphs and initiated the edited concept of fuzzy hypergraphs with certain isomorphism properties. Researchers are continuously working on the properties of fuzzy hypergraphs and their extensions including Hebbian structures of fuzzy hypergraphs [10], fuzzy coloring in fuzzy hypergraphs [11], transversals of fuzzy hypergraphs [12], certain properties of fuzzy hypergraphs [13], intuitionistic fuzzy hypergraphs [14, 15], bipolar fuzzy hypergraphs [16], various extensions of hypergraphs to deal with uncertainty [17], polar fuzzy hypergraphs [18, 19], and bipolar fuzzy soft hypergraphs [20].
All the existing approaches of hypergraphs based on fuzzy sets and their extensions can be applied using membership functions and parameterization tools. But, in some situations, when we have no additional information, membership functions, or parametric properties, the existing models based on hypergraphs are difficult to apply. Rough sets, introduced by Pawlak [21], are a key tool to handle such situations and study uncertain information without membership functions using upper and lower approximations. Rough sets are becoming a wide domain of research to study hybrid models based on graphs, relations, decision making problems, and fuzzy models, for instance, rough relations [22], rough graphs [23], rough fuzzy digraphs [24], fuzzy rough graphs [25], hybrid models based on rough sets, soft sets, and graphs [26], soft rough sets and rough soft sets [27], soft rough fuzzy sets [28], fuzzy sets combined with rough sets [29], rough set approximations for big data systems [30], hypergroups based on upper and lower approximations [31], modeling similarities in rough set theories [32], properties of certain types of rough relations [33], image classification based on rough sets [34], fuzzy rough feature selection based on graphs [35], risk minimization based in rough sets [36], fuzzy FCA (formal concept analysis) based on rough sets [37], applications of rough sets to graphs [38], hypergraphs [39], vertex rough graphs [40], FCA based on hypergraphs and rough sets [41], weak chromatic number of random hypergraphs [42], properties of totally balanced hypergraphs [43], Boolean operators based on rough sets [44], boundary optimization for rough sets [45] and connection between hypergroups, rough sets and hypergraphs [46], domination based rough sets [47], relationships between rough sets and topologies [48], planarity of product graphs in bipolar fuzzy environment [49], bipolar soft sets based on rough multipolar fuzzy approximations [50], and bipolar soft sets under rough Pythagorean fuzzy environment [51].
1.1. Motivation and Contribution
The motives of the present study are summarized as follows:(1)Graph theory has a wide range of applications in different domains to study pairwise relations among objects. But, in graphical models, usually certain information is ignored that two or more objects may satisfy common properties or characteristics. Hypergraph theory as a generalization of graph theory tackles this difficulty to study common characteristics of any collection of objects in a more efficient way.(2)In a hypergraph, binary values 0 and 1 are used to identify whether certain objects satisfy a common characteristic or not. Hypergraphs cannot study uncertain properties or partial belongingness of objects and their relations. Fuzzy hypergraphs and their extensions have been applied successfully to deal with uncertain information in hypergraphical models. But, all these existing approaches are based on additional suppositions and membership functions to compute the vagueness of objects. Rough sets are a power tool to discuss uncertainty using upper and lower approximations without any additional assumptions and predefined functions. Rough hypergraphs as an extension of hypergraphs can study incomplete information in hypergraphical models using given information, that is, no need for additional assumptions, which is the main focus of the present study.(3)Various hypergraphical structures have applications for map learning, link prediction, information entropy, etc. Hypergraphs are usually used to represent relations among social objects. Rough hypergraphs can be used to cope with uncertain relations among objects in social networks without suppositions of arbitrary membership values and functions unlike fuzzy hypergraphs and their extensions. Rough hypergraphs can be used in decision analysis for the grouping of different teams, storage of incompatible and flammable substances, and route-finding problems using distance measures.
The main contribution of this research paper is as follows:(1)This study proposes the novel concept of rough hypergraph using rough relations. A rough hypergraph is constructed on a set using equivalence relations.(2)The properties of isomorphism, conformality, linearity, duality, associativity, commutativity, distributivity, Helly property, and intersecting families in rough hypergraphs are studied in detail.(3)Certain operations on rough hypergraphs are discussed. The formulae for 2-section, L2-section, covering, coloring, rank, and antirank are established using approximation techniques.
1.2. Framework of the Paper
This paper is organized as follows:(1)Section 1 is based on the literature review and motives of the given study(2)Section 2 contains basic ideas, definitions, and terminologies from already existing articles that are used in the paper(3)Section 3 is the main focus of this research paper which contains novel concepts of certain types of rough hypergraphs and their interesting properties
2. Preliminaries
A hypergraph [1] on a nonempty set is written as a pair , where is a family of nonempty subsets of such that .
Definition 1 (see [21]). An approximation space on is a pair , where is defined as an equivalence (EQ) relation on . For any subset , the upper approximation and lower approximation of in are defined asHere, is known as EQ class of , and the pair is called a rough set.
Definition 2 (see [23]). A rough digraph on a nonvoid set is a 3-tuple such that(1) is an EQ relation on the vertex set (2)For , is a rough set on (3) is an EQ relation on any (4)For , is a rough relation on The rough digraph is also represented by the pair , where and are digraphs. If is an irreflexive and symmetric relation, and are simple graphs and is a rough graph on .
3. Rough Hypergraphs
In this section, the notion of a rough hypergraph is introduced with certain interesting properties of isomorphism, linearity, duality, and rough line graphs. We have discussed the 2-section, L2-section, rank, antirank, covering, and coloring of certain operations of rough hypergraphs.
Definition 3. Let be an EQ relation on and for ; let be a rough set on . Let be an EQ relation on , the power set of , such that for each , there exist iff .
Let be a family of nonempty subsets of ; then the upper and lower approximations and are defined asThe pair is known as a rough relation on . If , the power set of , then is a rough relation on .
Definition 4. A rough hypergraph on a nonempty set is a triplet such that(1) is an EQ relation on vertex set (2)For , is a rough set on (3) is an EQ relation on , that is, the family of nonempty subsets of (4)For , is a rough relation on ; that is, The rough hypergraph on is also denoted by the pair , where and are hypergraphs.
Example 1. Let be an EQ relation on as given in Figure 1. Let , then , and . Let be an EQ relation on as shown in Figure 2.
Let , then clearly , and D = . The rough hypergraph on is shown in Figure 3.
Definition 5. The degree of a vertex in a rough hypergraph is the sum of degrees of vertex in both hypergraphs and . It is denoted as , where and denote the number of hyperedges incident to in and , respectively.
The maximum degree of a rough hypergraph is denoted by and computed as the sum of maximum degrees in and , respectively; that is, .
The minimum degree of a rough hypergraph is denoted by and computed as the sum of minimum degrees in and , respectively; that is, .
Definition 6. The rank of a rough hypergraph is denoted by and defined as the sum of ranks of and ; that is, .
The antirank of a rough hypergraph is denoted by and defined as the sum of antiranks of and ; that is, .
A rough hypergraph is called a uniform rough hypergraph if . A uniform rough hypergraph is known as a - uniform rough hypergraph if .
Note that a rough hypergraph for which and is a rough graph. A 2-uniform rough hypergraph is a rough graph.
Definition 7. A rough hypergraph is called a partial rough hypergraph of a rough hypergraph if , , , and . It is written as .
A partial rough hypergraph is called induced if and contain all hyperedges of which has vertices from and , respectively.
Definition 8. A rough hyperpath of length between vertices and , denoted by , in a rough hypergraph is a sequence of distinct vertices and hyperedges , in both and . If , then the rough hyperpath is known as a rough hypercycle.
Definition 9. The distance between any two vertices and of a rough hypergraph is defined as the sum of lengths of shortest hyperpaths connecting and in both and ; that is, .
Definition 10. Let and be two rough hypergraphs on and , respectively. A homomorphism from into is a mapping if there exist homomorphisms and ; that is,(1)If is a hyperedge in , then is a hyperedge in (2)If is a hyperedge in , then is a hyperedge in A homomorphism which is a one-to-one correspondence between and is called an isomorphism. In this case, we say that the rough hypergraphs and are isomorphic to each other and write as .
Definition 11. The 2-section of a rough hypergraph is a rough graph with the same vertex set as in and two vertices are adjacent in and if they belong to the same hyperedge in and , respectively.
The L2-section of a rough hypergraph is the 2-section of with a pair of mappings , where and are such thatIn other words, the L2-section is a labeled 2-section of a rough hypergraph. As compared to 2-section, L2-section provides additional information to trace back the edges of which are associated with the hyperedges of . Thus, the original rough hypergraph can be easily constructed from the L2-section. The inverse is the rough hypergraph whose L2-section is with and .
Example 2. Consider a rough hypergraph shown in Figure 4. The 2-section of is given in Figure 4 with dashed lines and L2-section is given in Figure 5.
Remark 1. Let be a rough hypergraph; then, Definition 11 directly follows that(1)(2)
Lemma 1. Let and be two isomorphic rough hypergraphs; then, and .
Proof. Let and be two rough hypergraphs, then , and . The vertex set of and is the same for . Let ; then, there exists such that . Since , therefore is an isomorphism and is a hyperedge in such that , hence, in an edge in and so .
Let and = . It remains to show that the labeling functions and are equal. Clearly, for any , . Similarly, . Hence, the labeling functions and are equal and .
Definition 12. Let be a rough hypergraph on ; then, the distance between any two vertices and is defined aswhere and are lengths of shortest hyperpaths between and in and , respectively.
Lemma 2. Let be a rough hypergraph on ; then, for any .
Proof. If and are in different connected components in or , or both, then clearly and are in different components in or . In this case, . Assume that is a connected rough hypergraph; then, and are both connected hypergraphs and so is . Let be the shortest hyperpath in between vertices and . Then, by the construction of , there exists a hyperwalk in in . Let be the shortest path in . Clearly . Thus, corresponding to every , there exists such that , and so a hyperwalk of length in . A contradiction, hence . Similarly, . It clearly follows that .
We now study certain products of rough hypergraphs. In each product, the vertex set is the Cartesian product of the sets of vertices of all rough hypergraphs. The adjacency of edges is based on the adjacency properties defined in the product. Let denote any product of two rough hypergraphs and . For any rough hypergraph , if there exists another rough hypergraph such that , then is called the unit element. Note that must be a hypergraph with a single vertex and no loops. A rough hypergraph is called prime if whenever , then either or .
Definition 13. Let be a rough hypergraph on . The rough line graph of is a rough graph such that(1), where . That is, the hyperedge set of is the vertex set of . For any , if , then .(2), where . That is, the hyperedge set of is the vertex set of . For any , if , then .
Example 3. The rough line graph of Figure 3 is shown with dashed lines in Figure 6.
Definition 14. A rough hypergraph is called connected if and are both connected hypergraphs.
Lemma 3. A rough hypergraph is connected if and only if is a connected rough graph.
Definition 15. A rough hypergraph is called linear if and are linear hypergraphs, that is,(1)For any two hyperedges ,(a)(b)(2)For any two hyperedges ,(a)(b)
Theorem 1. Any nontrivial simple rough graph is a rough line graph of a linear rough hypergraph.
Proof. Let be a rough graph on . Assume, without loss of generality, that is a connected rough graph without multiple edges. A rough hypergraph can be constructed from as follows:(1)The vertex set of is the edge set of , that is, and (2)Let , then,(a)If is the collection of those edges of which has as incidence vertex, then is a hyperedge in ; that is, (b)If is the collection of those edges of which has as incidence vertex, then is a hyperedge in ; that is, It remains to show that is linear. Let and be two hyperedges in such that ; that is, both the edges and have two common vertices. Since has no multiple edges, therefore . Hence, is a linear hypergraph proving that is a linear rough hypergraph.
Theorem 2. For any rough hypergraph , .
Proof. Let be a rough hypergraph on and , , . The hyperedge set of is the vertex set of which is also the vertex set of . Let be the hyperedge set of such that ; then, is the edge set of and . Thus, . On the same argument, . Hence, .
Lemma 4. For any rough hypergraph ,(1)(2)If , then The proof of Lemma 4 is a direct consequence of Definitions 10 and 28.
Theorem 3. For any rough hypergraph , .
Proof. By Theorem 2 and Lemma 4, .
Theorem 4. If is a linear rough hypergraph, then = is also linear.
Proof. Since is linear, therefore and are linear hypergraphs. On the contrary, suppose that is not linear. Then, there exist hyperedges and in such that . Let . Definition 28 implies that and have two common vertices , that is, and . It denies the linearity of . Thus, is linear. Following similar arguments, the linearity of can be proved. Hence, is a linear rough hypergraph.
3.1. Cartesian Product
In this subsection, we introduce the concept of Cartesian product in rough hypergraphs and study its 2-section, L2-section, distance, covering, and coloring of the Cartesian product of rough hypergraphs.
Definition 16. Let = and = be two rough hypergraphs. The Cartesian product of and is a rough hypergraph which is defined as(1)(a)(b)(2)(a)(b)In short, and are the Cartesian products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the Cartesian product of hypergraphs, the Cartesian product of rough hypergraphs is associative, distributive with respect to the disjoint union, commutative, and a unit as a trivial rough hypergraph with a single vertex. That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4), where is a single vertex rough hypergraph without loops
Theorem 5. Let and be two rough hypergraphs; then,(1)(2)
Proof. Since, by Definition 16, , we first need to compute and . By Definition 6,Similarly, . Hence,By Definition 6, the antirank of is given asSimilarly, . Hence,
Lemma 5. Let and be two rough hypergraphs, then .
Proof. Since the vertex set of and is the same for , therefore the vertex set of and is the same. It only needs to show that the set of hyperedges of and is the same. By Definition 16, , where and . So, we haveAs the vertex set and is also same, so . Similarly, . Hence, .
Definition 17. Let and = be L2-sections of two hypergraphs and . The Cartesian product of L2-sections is a rough hypergraph with a labeling function , where and are defined as
Lemma 6. Let and be two rough hypergraphs, then(1)(2)
Proof. (1)It is clear from Lemma 7 that . It only needs to prove that the labeling function of and the function of Definition 17 are the same. Let be the L2-section of with a labeling function , where and are mappings. It is to be shown that and . By Definitions 11 and 17, Thus, , for each . Similarly, . Hence, which clearly proves .(2)For any rough hypergraph , is the rough hypergraph with a labeling function. The proof of part 2 is clear from part 1 and Remark 1. Using Remark 1, . By proof of part 1, . □
Theorem 6. Let and be two rough hypergraphs on and ; then, for any and ,
Proof. The proof of this theorem is a direct consequence of Proposition 5.1 of [5], Lemmas 2 and 6. Thus, for any two rough hypergraphs and ,Let be a rough hypergraph. If there exist prime rough hypergraphs such that , then it is called PFD (prime factor decomposition) of into factors w.r.t the Cartesian product.
Remark 2. Every connected rough hypergraph has a unique PFD with respect to the Cartesian product.
The method to obtain a PFD of a rough hypergraph using its L2-section is illustrated in Algorithm 1.
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Definition 18. Let and be two rough hypergraphs on and , respectively. A homomorphism from into is a mapping if there exist homomorphisms and ; that is,(1)If is a hyperedge in , then is a hyperedge in (2)If is a hyperedge in , then is a hyperedge in A homomorphism which is a one-to-one correspondence between and is called an isomorphism. In this case, we say that the rough hypergraphs and are isomorphic to each other and write as .
Definition 19. Let and be two rough hypergraphs on and , respectively. A surjective homomorphism is called a covering projection if(1), for all , (2), for all , (3), for all distinct , (4), for all distinct , The rough hypergraph is called a -fold covering of , and is called quotient rough hypergraph of . If , is called a double cover of .
Definition 20. Let be a rough hypergraph on . The sets and are called independent if they contain no hyperedge of and , respectively. The cardinalities of the largest independent sets are denoted by and and are called the independence number of and , respectively. The value is called independence number of .
Definition 21. Let be a rough hypergraph on . The subsets and are called covers of and , respectively, if and , for each and . The cardinalities of minimal covers are denoted by and and are called covering numbers of and , respectively. The average value is called covering number of .
Definition 22. Let be a rough hypergraph on . The fractional covers of and are, respectively, the mappings and such thatThe value is called fractional covering number (FC number) of .
Definition 23. Let be a rough hypergraph on . The subsets and are called matching if every pair of hyperedges from and are mutually disjoint. The cardinalities of maximal matchings are denoted by and and are called matching numbers of and , respectively. The matching number of is computed as
Definition 24. Let be a rough hypergraph on . The minimum number of mutually disjoint hyperedges whose union is the sets of vertices and is called the partition number of and , respectively, denoted by and . The value is called the partition number of . If such partitions do not exist, then .
We now study certain products of rough hypergraphs. In each product, the vertex set is the Cartesian product of the sets of vertices of all rough hypergraphs. The adjacency of edges is based on the adjacency properties defined in the product. Let denote any product of two rough hypergraphs and . For any rough hypergraph , if there exists another rough hypergraph such that , then is called the unit element. Note that must be a hypergraph with a single vertex and no loops.
Definition 25. A rough hypergraph is called conformal if and are both conformal hypergraphs. That is, corresponding to each clique of 2-section (and ), there is a hyperedge in (and ).
Definition 26. Let be a rough hypergraph. The collection of all hyperedges in (and ) containing a common vertex is called a star of (and ), denoted by (and ). The pair is called a rough star of . The subsets and are called intersecting families of and if every pair of hyperedges of and have nonempty intersection. The pair is called a rough intersecting family of . A rough hypergraph is said to satisfy Helly property if each rough intersecting family in is a rough star.
Definition 27. Let and be -fold and -fold coverings of rough hypergraphs and via covering projections and , respectively; then, the Cartesian product is defined as(1), for all (2), for all (3), for all , (4), for all , (5), for all , (6), for all ,
Theorem 7. Let and be -fold and -fold coverings of rough hypergraphs and via covering projections and , respectively; then, is a -fold covering of via covering projection .
Proof. The mapping is given in Definition 27. We first need to show that is a surjective homomorphism. Let be a hyperedge in ; then, . Since and are homomorphisms, therefore and is a hyperedge in . Thus, is a hyperedge in . Similarly, is a hyperedge and ; are hyperedges in showing that is a homomorphism. The surjectivity of and is obvious from the surjectivity of , , , and .
Let be a vertex in ; then,Similarly,Consider a hyperedge in ; then,Similarly,Let be a hyperedge in such that and . It follows that , a contradiction. Similarly, we can prove the other cases. Hence, is covering projection and is a -fold covering of . □
Theorem 8. Let and be two rough hypergraphs; then, is conformal if and only if and are conformal.
Proof. Let and be two conformal rough hypergraphs. Let be a rough clique in ; then, by Lemma 7, there are two possibilities.
Case 1. There exists a rough clique in such that , for some , and , for some . Since is a rough clique, therefore, is also a rough clique. As is a conformal rough hypergraph, therefore there exist hyperedges and corresponding to and in and . Thus, is a rough hyperedge corresponding to showing that is a conformal rough hypergraph.
Case 2. There exists a rough clique in such that , for some , and , for some . This case can be proved along the same lines as Case 1. Hence, is a conformal rough hypergraph.
Conversely, let be a conformal rough hypergraph. Let and be rough cliques in and , respectively. By Lemma 7, is a rough clique in . Since is a conformal rough hypergraph, therefore, there exists a rough hyperedge in corresponding to . By Lemma 7, there exist rough hyperedges and in and such that . Clearly, and correspond to in and , respectively, proving that are conformal rough hypergraphs. □
Theorem 9. Let and be two rough hypergraphs; then, has the Helly property iff and have the Helly property.
Definition 28. The dual of a rough hypergraph is a rough hypergraph , where(1)The hyperedge set of is the vertex set of , that is, (2)The edge set of is the vertex set of , that is, (3)If , then is the hyperedge set of such that , that is, is the collection of those hyperedges of which share the common vertex (4)If , then is the hyperedge set of such that , that is, is the collection of those hyperedges of which share the common vertices
Remark 3. Let and be two rough hypergraphs; then, may not be equal to . Since, for any two hypergraphs and , is not equal to in general, therefore, the equality also does not hold in the case of rough hypergraphs because a rough hypergraph contains two hypergraphs as upper and lower approximations. We discuss this fact using an example of two hypergraphs shown in Figures 7 and 8.
It is easy to check that has nine edges and so has nine vertices. But, has three vertices. The vertex sets of and are not equal and it proves our claim.
3.2. Square Product
In this subsection, we introduce the concept of square product in rough hypergraphs and discuss its associativity, commutativity, distributivity, 2-section, rank, and antirank properties.
Definition 29. Let be any product of rough hypergraphs and denotes the vertex set of , for any . The mapping is called the projection of onto factor , where and are the projection mappings defined as
Definition 30. Let and be two rough hypergraphs. The square product of and is a rough hypergraph which is defined as(1)(a)(b)(2)(a)(b)In short, and are the square products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the square product of hypergraphs, the square product of rough hypergraphs is associative, distributive with respect to the disjoint union, commutative, and a unit as a trivial hypergraph with a single vertex such that . That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4), where is a single vertex hypergraph without loops(5)The projections and are homomorphisms
Definition 31. Let be a rough hypergraph; then, is called an -uniform rough hypergraph if and are both -uniform hypergraphs; that is, for each and , .
Lemma 7. Let and be two -uniform rough hypergraphs; then, .
Proof. Since the vertex set of and is the same for , therefore, the vertex set of and is the same. It only needs to show that the set of hyperedges of and is the same. By Definition 30, , where and . As and are -uniform rough hypergraphs, so we haveAs the vertex set and is also the same, so . Similarly, . Hence, .
Theorem 10. Let and be two rough hypergraphs; then,(1)(2)
Proof. Since, by Definition 30, , we first need to compute and . By Definition 6,Similarly, . Hence,Using Definition 6, the antirank of is given asSimilarly, . Hence,
3.3. Direct Product
In this section, we introduce the extension of the concept of the square product to direct product of rough hypergraphs and discuss its associativity, commutativity, distributivity, 2-section, rank, and antirank properties.
Definition 32. Let and be two rough hypergraphs. The direct product of and is a rough hypergraph which is defined as(1)(a)(b)(2)(a)(b)In short, and are the MRP direct products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the MRP direct product of hypergraphs, the MRP direct product of rough hypergraphs is associative, right distributive with respect to the disjoint union, commutative, and a unit as a trivial hypergraph with a single vertex such that . That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4), where is a single vertex hypergraph without loops(5)The projections and may not be weak homomorphisms
Lemma 8. Let and be two rough hypergraphs; then, .
Proof. Since the vertex set of and is the same for , therefore, the vertex set of and the union of and is the same. It only needs to show that the set of hyperedges of and is the same. By Definition 32, , where and . So, we haveThus, . Similarly, . Hence, .
Theorem 11. Let and be two rough hypergraphs; then,(1)(2)
Proof. Since, by Definition 32, , we first need to compute and . By Definition 6,Similarly, . Hence,Using Definition 6, the antirank of is given asSimilarly, . Hence,
3.4. Union and Intersection
In this subsection, we introduce the concepts union of the intersection of rough hypergraphs and study their properties.
Definition 33. Let and be two rough hypergraphs. The union f and is a rough hypergraph , where and .
The intersection f and is a rough hypergraph , where and .
In short, and are the union and intersection of lower approximate hypergraphs and . Similarly, for the upper approximate hypergraphs, just like the union (intersection) of hypergraphs, the union (intersection) of rough hypergraphs is associative, commutative, and distributive. That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4)(5)(6)The projections and may not be weak homomorphisms(7)The projections and may not be weak homomorphisms
Remark 4. Let and be two rough hypergraphs, then(1)(2)(3)(4)
3.5. Strong Product
In this subsection, we introduce the concept of a strong product using the Cartesian and square product of rough hypergraphs. We illustrate the notions of associativity, commutativity, distributivity, 2-section, distance, rank, and antirank properties of the strong product of rough hypergraphs.
Definition 34. Let and be two rough hypergraphs. The strong product of and is a rough hypergraph , where and . In other words, the strong product of and is the union of Cartesian product and square product of rough hypergraphs and .
In short, and are the strong products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the strong product of hypergraphs, the strong product of rough hypergraphs is associative, right distributive with respect to the disjoint union, commutative, and a unit as a trivial hypergraph with a single vertex such that . That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4), where is a single vertex hypergraph without loops(5)The projections and may not be weak homomorphisms(6)(7)(8)
Theorem 12. Let and be two rough hypergraphs on and ; then, for any and ,
Proof. The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2, and the result . Thus, for any two rough hypergraphs and ,Similarly, , and the result follows.
3.6. Normal Product
In this subsection, we introduce the concept of a normal product using the Cartesian and direct product of rough hypergraphs. We elaborate on the notions of associativity, commutativity, distributivity, 2-section, distance, rank, and antirank properties of the normal product of rough hypergraphs.
Definition 35. Let and be two rough hypergraphs. The strong product of and is a rough hypergraph , where and . In other words, the strong product of and is the union of Cartesian product and direct product of rough hypergraphs and .
In short, and are the strong products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the strong product of hypergraphs, the strong product of rough hypergraphs is associative, right distributive with respect to the disjoint union, commutative, and a unit as a trivial hypergraph with a single vertex such that . That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4), where is a single vertex hypergraph without loops(5)The projections and may not be weak homomorphisms(6)(7)(8)
Theorem 13. Let and be two rough hypergraphs on and , then for any and ,
Proof. The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2 and the result . Thus, for any two rough hypergraphs and . □
3.7. Lexicographic Product and Costrong Product
In this subsection, we describe the properties of the lexicographic product and costrong product of rough hypergraphs.
Definition 36. Let and be two rough hypergraphs. The lexicographic product of and is a rough hypergraph which is defined as(1)(a)(b)(2)(a)(b)In short, and are the lexicographic products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the lexicographic product of hypergraphs, the lexicographic product of rough hypergraphs is associative, right distributive with respect to the disjoint union, noncommutative, and a unit (left identity) as a trivial hypergraph with a single vertex such that . That is, for any rough hypergraphs , , and , the following properties hold:(1)(2)(3)(4), where is a single vertex hypergraph without loops(5)(6)The projections and may not be weak homomorphisms(7)(8)(9)
Theorem 14. Let and be two rough hypergraphs on and ; then, for any and ,
Proof. The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2, and the result . Thus, for any two rough hypergraphs and ,Similarly for upper approximate hypergraphs, the result follows.
Definition 37. Let and be two rough hypergraphs. The costrong product of and is a rough hypergraph which is defined as .
In short, and are the costrong products of lower approximate hypergraphs and upper approximate hypergraphs , respectively. Just like the costrong product of hypergraphs, the costrong product of rough hypergraphs is associative, right distributive with respect to the disjoint union, commutative, and a unit (left identity) as a trivial hypergraph with a single vertex such that .
Remark 5. Let and be two rough hypergraphs, then(1)(2)(3)
3.8. Limitations of the Proposed Study
Apart from all the benefits, rough hypergraphs also have some shortcomings and disadvantages. Rough sets and hypergraphs are both complex mathematical structures and are not simple to apply for the given information. The computation of rough relations using power sets is a lengthy and tricky task. There are a lot of complicated calculations which make it difficult to study hypergraphical structures using rough sets. The calculation complexity not only increases time consumption but also increases the probability of errors.
4. Conclusions and Future Directions
Rough models combined with other algebraic structures retain the property to study uncertain and vague information using approximation techniques. To discuss approximate relations among more than two objects, rough graphs cannot give error-free results. In this research paper, the notion of a rough set was applied to hypergraphs to introduce the novel concept of rough hypergraphs. Certain important properties of isomorphism, conformality, linearity, duality, associativity, commutativity, distributivity, Helly property, and intersecting families of rough hypergraphs are illustrated in detail. The formulae of distance function, 2-section, L2-section, covering, coloring, rank, and antirank of certain products of rough hypergraphs are established in terms of corresponding rough hypergraphs. This work can further be extended to (1) Dombi fuzzy rough hypergraphs, (2) bipolar fuzzy rough hypergraphs, and (3) picture fuzzy rough hypergraphs.
Data Availability
No data were used to support this study.
Ethical Approval
This article does not contain any studies with human participants or animals performed by the author.
Conflicts of Interest
The author declares she has no conflicts of interest regarding the publication of this research article.