# A Theoretical Investigation Based on the Rough Approximations of Hypergraphs

Authors: Musavarah Sarwar
Journal: Journal of Mathematics
 (2022)

---

## 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.


---

## Body

## 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], m−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], l2−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)
Section1 is based on the literature review and motives of the given study(2)
Section2 contains basic ideas, definitions, and terminologies from already existing articles that are used in the paper(3)
Section3 is the main focus of this research paper which contains novel concepts of certain types of rough hypergraphs and their interesting properties


## 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)
Section1 is based on the literature review and motives of the given study(2)
Section2 contains basic ideas, definitions, and terminologies from already existing articles that are used in the paper(3)
Section3 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 Q is written as a pair H=Q,D, where D=D1,D2,…,Dr is a family of nonempty subsets of Q such that ∪k=1rDk=Q.Definition 1 (see [21]).
An approximation space onT is a pair Q,φ, where φ is defined as an equivalence (EQ) relation on T. For any subset A⊆Q, the upper approximation φ¯A and lower approximation φ¯A of A in Q,φ are defined as(1)φ¯A=d∈Q|dφ∩A≠∅,φ¯A=d∈Q|dφ⊆A.
Here,dφ=g∈Q|d,g∈φ is known as EQ class of d∈T, and the pair φ¯A,φ¯A is called a rough set.Definition 2 (see [23]).
A rough digraph on a nonvoid setQ is a 3-tuple G=Q,φA,ψD such that(1)
φ is an EQ relation on the vertex set Q(2)
ForA⊆Q, φA=φ¯A,φ¯A is a rough set on Q(3)
ψD is an EQ relation on any M⊆Q×Q(4)
ForD⊆A×A, ψD=ψ¯D,ψ¯D is a rough relation on φA=φ¯A,φ¯A
The rough digraph is also represented by the pairU=U¯,U¯, where U¯=φ¯A,ψ¯D and U¯=φ¯A,ψ¯D are digraphs. If D is an irreflexive and symmetric relation, U¯ and U¯ are simple graphs and U is a rough graph on Q.


## 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.
Letφ be an EQ relation on Q and for A⊆Q; let φ¯A,φ¯A be a rough set on Q. Let ψ be an EQ relation on ℳ⊆PQ∖∅, the power set of Q, such that for each d1,g1∈U1, there exist d2,g2∈U2 iff U1,U2∈ψ.
LetD⊆PA∖∅ be a family of nonempty subsets of A; then the upper and lower approximations ψ¯D and ψ¯D are defined as(2)ψ¯D=V∈ℳ|Vψ∩D≠∅,ψ¯D=V∈ℳ|Vψ⊆D.
The pairψ¯D,ψ¯D is known as a rough relation on Q. If ψ¯D⊆Pφ¯A, the power set of φ¯A, then ψ¯D,ψ¯D is a rough relation on φ¯A,φ¯A.Definition 4.
A rough hypergraph on a nonempty setQ is a triplet Q,φA,ψD such that(1)
φ is an EQ relation on vertex set Q(2)
ForA⊆Q, φ¯A,φ¯A is a rough set on Q(3)
ψ is an EQ relation on ℳ⊆PQ∖∅, that is, the family of nonempty subsets of Q(4)
ForD⊆PA∖∅, ψ¯D,ψ¯D is a rough relation on φ¯A,φ¯A; that is, ψ¯D⊆Pφ¯A
The rough hypergraph onQ is also denoted by the pair ℛ=ℛ¯,ℛ¯, where ℛ¯=φ¯A,ψ¯D and ℛ¯=φ¯A,ψ¯D are hypergraphs.Example 1.
Letφ be an EQ relation on Q=o,d,t,g,y,h as given in Figure 1. Let A=o,t,g,d,h, then φ¯A=o,h,d,t, and φ¯A=o,d,t,g,y,h. Let ψ be an EQ relation on ℳ⊆PQ∖∅ as shown in Figure 2.
LetD=o,g,h,o,t,d,y,o,t,d,g, then clearly ψ¯D=o,t,d,y,o,t,d,g, and ψ¯ D = o,g,h,o,y,h,o,g,y,h,o,t,d,g,o,t,d,y. The rough hypergraph on Q is shown in Figure 3.Figure 1
Approximation spaceQ,φ.Figure 2
Approximation spaceℳ,ψ.Figure 3
Rough hypergraphφ¯A,ψ¯D,φ¯A,ψ¯D.Definition 5.
The degree of a vertexk in a rough hypergraph ℛ=ℛ¯,ℛ¯ is the sum of degrees of vertex k in both hypergraphs ℛ¯=ψ¯A,ψ¯D and ℛ¯=ψ¯A,ψ¯D. It is denoted as degk=1/2deg¯k+deg¯k, where deg¯k and deg¯k denote the number of hyperedges incident to k in ℛ¯k and ℛ¯k, respectively.
The maximum degree of a rough hypergraph is denoted byΔℛ and computed as the sum of maximum degrees in ℛ¯ and ℛ¯, respectively; that is, Δℛ=1/2Δℛ¯+Δℛ¯=1/2maxk∈Pdeg¯k+maxk∈Pdeg¯k.
The minimum degree of a rough hypergraph is denoted byδℛ and computed as the sum of minimum degrees in ℛ¯ and ℛ¯, respectively; that is, δℛ=1/2δℛ¯+δℛ¯=1/2maxk∈Pdeg¯k+mink∈Pdeg¯k.Definition 6.
The rank of a rough hypergraph is denoted byrℛ and defined as the sum of ranks of ℛ¯ and ℛ¯; that is, rℛ=1/2rℛ¯+rℛ¯=1/2maxE∈ψ¯DE+maxE∈ψ¯DE.
The antirank of a rough hypergraph is denoted bysℛ and defined as the sum of antiranks of ℛ¯ and ℛ¯; that is, sℛ=sℛ¯+sℛ¯=1/2maxk∈PminE∈ψ¯DE+minE∈ψ¯DE.
A rough hypergraph is called a uniform rough hypergraph ifrℛ=sℛ. A uniform rough hypergraph is known as a p- uniform rough hypergraph if rℛ¯=rℛ¯=p.
Note that a rough hypergraph for whichrℛ¯≤2 and rℛ¯≤2 is a rough graph. A 2-uniform rough hypergraph is a rough graph.Definition 7.
A rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is called a partial rough hypergraph of a rough hypergraph S=φ¯B,ψ¯F,φ¯B,ψ¯F if φ¯A⊆φ¯B, ψ¯D⊆ψ¯F, φ¯A⊆φ¯B, and ψ¯D⊆ψ¯F. It is written as ℛ⊆S.
A partial rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is called induced if ψ¯F and ψ¯F contain all hyperedges of S which has vertices from φ¯B and φ¯B, respectively.Definition 8.
A rough hyperpath of lengthq between vertices g and k, denoted by ko−−kq, in a rough hypergraph ℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is a sequence of distinct vertices and hyperedges ko,D1,k1,D2,k2,D3, …,Dq−1,kq−1,Dq,kq in both ℛ¯=φ¯A,ψ¯D and ℛ¯=φ¯A,ψ¯D. If ko=kq, then the rough hyperpath is known as a rough hypercycle.Definition 9.
The distancedℛk,g between any two vertices k and g of a rough hypergraph ℛ is defined as the sum of lengths of shortest hyperpaths connecting k and g in both ℛ¯ and ℛ¯; that is, dℛk,g=1/2dℛ¯k,g+dℛ¯k,g.Definition 10.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2, respectively. A homomorphism from ℛ1 into ℛ2 is a mapping f:Q1⟶Q2 if there exist homomorphisms f¯:Q1⟶Q2 and f¯:Q1⟶Q2; that is,(1)
Iff¯D is a hyperedge in ℛ2¯, then D is a hyperedge in ℛ1¯(2)
Iff¯D is a hyperedge in ℛ2¯, then D is a hyperedge in ℛ1¯
A homomorphism which is a one-to-one correspondence betweenQ1 and Q2 is called an isomorphism. In this case, we say that the rough hypergraphs ℛ1 and ℛ2 are isomorphic to each other and write as ℛ1≅ℛ2.Definition 11.
The 2-sectionℛ2=ℛ¯2,ℛ¯2=φ¯A,ψ¯D2,φ¯A,ψ¯D2 of a rough hypergraph ℛ=ℛ¯,ℛ¯ is a rough graph with the same vertex set as in ℛ and two vertices are adjacent in ℛ¯2 and ℛ¯2 if they belong to the same hyperedge in ℛ¯ and ℛ¯, respectively.
The L2-sectionℛL2=ℛ¯L¯2,ℛ¯L¯2=φ¯A,ψ¯D2,L¯,φ¯A,ψ¯D2,L¯ of a rough hypergraph ℛ=ℛ¯,ℛ¯ is the 2-section of ℛ with a pair of mappings L=L¯,L¯, where L¯:ψ¯D2⟶ψ¯D and L¯:ψ¯D2⟶ψ¯D are such that(3)L¯kg=E¯∈ψ¯D|k,g∈E¯,L¯kg=E¯∈ψ¯D|k,g∈E¯.
In 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 ofℛ2 which are associated with the hyperedges of ℛ. Thus, the original rough hypergraph can be easily constructed from the L2-section. The inverse ℛL2−1 is the rough hypergraph whose L2-section is ℛL2 with ψ¯D=∪kg∈ψ¯D2L¯kg and ψ¯D=∪kg∈ψ¯D2L¯kg.Example 2.
Consider a rough hypergraphℛ shown in Figure 4. The 2-section ℛ2 of ℛ is given in Figure 4 with dashed lines and L2-section ℛL2 is given in Figure 5.Figure 4
2-section of the rough hypergraphφ¯A,ψ¯D,φ¯A,ψ¯D.Figure 5
L2-section of rough hypergraphφ¯A,ψ¯D,φ¯A,ψ¯D.Remark 1.
Letℛ be a rough hypergraph; then, Definition 11 directly follows that(1)
ℛ2−1=ℛ(2)
ℛ2−12=ℛ2Lemma 1.
Letℛ1 and ℛ2 be two isomorphic rough hypergraphs; then, ℛ12≅ℛ22 and ℛ1L2≅ℛ2L2.Proof.
Letℛ1=φ¯A1,ψ¯D1,φ¯A1,ψ¯D1 and ℛ2=φ¯A2,ψ¯D2,φ¯A2,ψ¯D2 be two rough hypergraphs, then ℛ12=φ¯A1,ψ¯D12,φ¯A1,ψ¯D12, and ℛ22=φ¯A2,ψ¯D22,φ¯A2,ψ¯D22. The vertex set of ℛj and ℛj2 is the same for j=1,2. Let kg∈ψ¯D12; then, there exists E¯∈ψ¯D1 such that k,g⊆E¯. Since ℛ1≅ℛ2, therefore f:ℛ1⟶ℛ2 is an isomorphism and fE¯ is a hyperedge in ℛ2 such that fk,fg∈fE¯, hence, fkfg in an edge in ℛ22 and so R12≅R22.
Letℛ1L2=φ¯A1,ψ¯D12,L¯1,φ¯A1,ψ¯D12,L¯1 and ℛ2L2 =  φ¯A2,ψ¯D22,L¯2,φ¯A2,ψ¯D22,L¯2. It remains to show that the labeling functions L1=L¯1,L¯1 and L2=L¯2,L¯2 are equal. Clearly, for any k1g1∈ψ¯D12, L¯1=E¯1|k1,g1∈E¯1=fE|fk,fg∈fE¯=L¯2. Similarly, L¯1=L¯2. Hence, the labeling functions L1 and L2 are equal and ℛ1L2≅ℛ2L2.Definition 12.
Letℛ be a rough hypergraph on Q; then, the distance between any two vertices k and g is defined as(4)dℛk,g=12dℛ¯k,g+dℛ¯k,g,where dℛ¯k,g and dℛ¯k,g are lengths of shortest hyperpaths between k and g in ℛ¯ and ℛ¯, respectively.Lemma 2.
Letℛ be a rough hypergraph on Q; then, for any k,g∈Qdℛk,g=dℛ2k,g.Proof.
Ifk and g are in different connected components in ℛ¯ or ℛ¯, or both, then clearly k and g are in different components in ℛ¯2 or ℛ¯2. In this case, ℛ¯=ℛ¯=∞. Assume that ℛ is a connected rough hypergraph; then, ℛ¯ and ℛ¯ are both connected hypergraphs and so is ℛ2. Let k,E¯1,k1,E¯2,…,ks−1,E¯s,g be the shortest hyperpath in ℛ¯ between vertices k and g. Then, by the construction of ℛ¯2, there exists a hyperwalk in k,e¯1,k1,e¯2,…,ks−1,e¯s,ks=g in ℛ¯2. Let k,f¯1,k1,f¯2,…,kp−1,f¯p,g be the shortest path in ℛ¯. Clearly s≥p. Thus, corresponding to every f¯j∈ψ¯D2, there exists F¯j∈ψ¯D such that f¯j⊆F¯j, and so a hyperwalk of length p in ℛ¯. A contradiction, hence dℛ¯k,g=dℛ¯2k,g. Similarly, dℛ¯k,g=dℛ¯2k,g. It clearly follows that dℛk,g=dℛ2k,g.
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. Letℛ1⊛ℛ2 denote any product of two rough hypergraphs ℛ1 and ℛ2. For any rough hypergraph ℛ, if there exists another rough hypergraph U such that ℛ⊛U=ℛ, then U is called the unit element. Note that U must be a hypergraph with a single vertex and no loops. A rough hypergraph ℛ is called prime if whenever ℛ=ℛ1⊛ℛ2, then either ℛ1=U or ℛ2=U.Definition 13.
Letℛ be a rough hypergraph on Q. The rough line graph of ℛ is a rough graph Lℛ=Lℛ¯,Lℛ¯ such that(1)
LJ¯=φ¯AL,ψ¯DL, where φ¯AL=ψ¯D. That is, the hyperedge set of ℛ¯=φ¯A,ψ¯D is the vertex set of Lℛ¯. For any Ei,Ek∈ψ¯D, if Ei∩Ek≠∅, then EiEk∈ψ¯DL.(2)
LJ¯=φ¯AL,ψ¯DL, where φ¯AL=ψ¯D. That is, the hyperedge set of ℛ¯=φ¯A,ψ¯D is the vertex set of Lℛ¯. For any Ei,Ek∈ψ¯D, if Ei∩Ek≠∅, then EiEk∈ψ¯DL.Example 3.
The rough line graphLℛ of Figure 3 is shown with dashed lines in Figure 6.Figure 6
Rough line graphLℛ.Definition 14.
A rough hypergraphℛ is called connected if ℛ¯ and ℛ¯ are both connected hypergraphs.Lemma 3.
A rough hypergraphℛ is connected if and only if Lℛ is a connected rough graph.Definition 15.
A rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is called linear if ℛ¯=φ¯A,ψ¯D and ℛ¯=φ¯A,ψ¯D are linear hypergraphs, that is,(1)
For any two hyperedgesE¯i,E¯j∈ψ¯D,(a)
E¯i⊆E¯j⇒i=j(b)
E¯i∩E¯j≤1(2)
For any two hyperedgesE¯i,E¯j∈ψ¯D,(a)
E¯i⊆E¯j⇒i=j(b)
E¯i∩E¯j≤1Theorem 1.
Any nontrivial simple rough graph is a rough line graph of a linear rough hypergraph.Proof.
LetG=φ¯C,ψ¯B,φ¯C,ψ¯B be a rough graph on Q. Assume, without loss of generality, that U is a connected rough graph without multiple edges. A rough hypergraph ℛ=α¯,β¯,α¯,β¯ can be constructed from U as follows:(1)
The vertex set ofℛ is the edge set of G, that is, α¯=ψ¯B and α¯=ψ¯B(2)
LetQ=k1,k2,…,km, then,(a)
IfE¯i is the collection of those edges of φ¯C,ψ¯B which has ki as incidence vertex, then E¯i∈β¯ is a hyperedge in α¯,β¯; that is, E¯i=kikj∈ψ¯B|kj∈Q,j=1,2,3,…(b)
IfE¯i is the collection of those edges of φ¯C,ψ¯B which has ki as incidence vertex, then E¯i∈β¯ is a hyperedge in α¯,β¯; that is, E¯i=kikj∈ψ¯B|kj∈Q,j=1,2,3,…
It remains to show thatℛ is linear. Let E¯i and E¯j be two hyperedges in α¯,β¯ such that E¯i∩E¯j=er,et; that is, both the edges er and et have two common vertices. Since φ¯C,ψ¯B has no multiple edges, therefore er=et. Hence, α¯,β¯ is a linear hypergraph proving that ℛ is a linear rough hypergraph.Theorem 2.
For any rough hypergraphℛ, Lℛ≅ℛ∗2.Proof.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q=k1,k2,…,kn and ψ¯D=E¯1,E¯2,…, E¯m, ψ¯D=E¯1,E¯2,…,E¯m. The hyperedge set of ℛ is the vertex set of Lℛ which is also the vertex set of ℛ∗2. Let X¯1,X¯2,…,X¯n be the hyperedge set of ℛ¯∗ such that X¯i=E¯j∣ki∈E¯j; then, ElEj|El,Ej∈Xi is the edge set of Lℛ¯ and ℛ¯∗2. Thus, Lℛ¯≅ℛ¯∗2. On the same argument, Lℛ¯≅ℛ¯∗2. Hence, Lℛ≅ℛ∗2.Lemma 4.
For any rough hypergraphℛ,(1)
ℛ∗∗≅ℛ(2)
Ifℛ1≅ℛ2, then ℛ1∗≅ℛ2∗
The proof of Lemma4 is a direct consequence of Definitions 10 and 28.Theorem 3.
For any rough hypergraphℛ, Lℛ∗≅ℛ2.Proof.
By Theorem2 and Lemma 4, ℛ2≅ℛ∗∗2≅Lℛ∗.Theorem 4.
Ifℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is a linear rough hypergraph, then ℛ∗ =  φ¯A∗,ψ¯D∗,φ¯A∗,ψ¯D∗ is also linear.Proof.
Sinceℛ is linear, therefore φ¯A,ψ¯D and φ¯A,ψ¯D are linear hypergraphs. On the contrary, suppose that φ¯A∗,ψ¯D∗ is not linear. Then, there exist hyperedges X¯i and X¯j in φ¯A∗,ψ¯D∗ such that X¯i∩X¯j=2. Let X¯i∩X¯j=Et,Es. Definition 28 implies that Et and Es have two common vertices ki,kj, that is, ki,kj∈Et and ki,kj∈Es. It denies the linearity of ℛ¯. Thus, J¯∗ is linear. Following similar arguments, the linearity of J¯∗ can be proved. Hence, J∗ 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.
Letℛ1 =  ℛ¯1,ℛ¯1 and ℛ2 =  ℛ¯2,ℛ¯2 be two rough hypergraphs. The Cartesian product of ℛ1 and ℛ2 is a rough hypergraph ℛ1□ℛ2=ℛ¯1□ℛ¯2,ℛ¯1□ℛ¯2 which is defined as(1)
ℛ¯1□ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1□ψ¯D2=k1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2(2)
ℛ¯1□ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1□ψ¯D2=k1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2
In short,ℛ¯1□ℛ¯2 and ℛ¯1□ℛ¯2 are the Cartesian products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1□ℛ2□ℛ3≅ℛ1□ℛ2□ℛ3(2)
ℛ1□ℛ2≅ℛ2□ℛ1(3)
ℛ1□ℛ2∪ℛ3≅ℛ1□ℛ2∪ℛ1□ℛ2(4)
ℛ□U≅ℛ, where U is a single vertex rough hypergraph without loopsTheorem 5.
Letℛ1 and ℛ2 be two rough hypergraphs; then,(1)
rℛ1□ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(2)
sℛ1□ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Proof.
Since, by Definition16, ℛ1□ℛ2=ℛ¯1□ℛ¯2,ℛ¯1□ℛ¯2, we first need to compute rℛ¯1□ℛ¯2 and rℛ¯1□ℛ¯2. By Definition 6,(5)rℛ¯1□ℛ¯2=maxE∈ψ¯D1□ψ¯D2E=maxk1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2=maxE¯2|E¯2∈ψ¯D2∪E¯1|E¯1∈ψ¯D1=maxmaxE¯1∈ψ¯D1E¯1,maxE¯2∈ψ¯D2E¯2⇒rℛ¯1□ℛ¯2=maxrℛ¯1,rℛ¯2.
Similarly,rℛ¯1□ℛ¯2=maxrℛ¯1,rℛ¯2. Hence,(6)rℛ1□ℛ2=12maxrℛ¯1,rℛ¯2+maxℛ¯1,rℛ¯2=maxrℛ¯1+rℛ¯12,rℛ¯2+rℛ¯22,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12=maxrℛ1,rℛ2,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12.
By Definition6, the antirank of ℛ1□ℛ2 is given as(7)sℛ¯1□ℛ¯2=minE∈ψ¯D1□ψ¯D2E=mink1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2=minE¯2|E¯2∈ψ¯D2∪E¯1|E¯1∈ψ¯D1=minminE¯1∈ψ¯D1E¯1,minE¯2∈ψ¯D2E¯2⇒sℛ¯1□ℛ¯2=minsℛ¯1,sℛ¯2.
Similarly,sℛ¯1□ℛ¯2=minsℛ¯1,sℛ¯2. Hence,(8)sℛ1□ℛ2=12minsℛ¯1,sℛ¯2+minsℛ¯1,sℛ¯2=minsℛ¯1+sℛ¯12,sℛ¯2+sℛ¯22,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12=minsℛ1,sℛ2,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12.Lemma 5.
Letℛ1 and ℛ2 be two rough hypergraphs, then ℛ1□ℛ22=ℛ12□ℛ22.Proof.
Since the vertex set ofℛi and ℛi2 is the same for i=1,2, therefore the vertex set of ℛ1□ℛ22 and ℛ12□ℛ22 is the same. It only needs to show that the set of hyperedges of ℛ1□ℛ22 and ℛ12□ℛ22 is the same. By Definition 16, ℛ1□ℛ22=ℛ¯1□ℛ¯22,ℛ¯1□ℛ¯22, where ℛ¯1□ℛ¯22=φ¯A1×φ¯A2,ψ¯D1□ψ¯D22 and ℛ¯1□ℛ¯22=φ¯A1×φ¯A2,ψ¯D1□ψ¯D22. So, we have(9)ψ¯D1□ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1□ψ¯D2=x1,x2x1,y2|y1=x1,x2,y2⊆E¯2∈ψ¯D2∪x1,x2y1,x2|y2=x2,x1,y1⊆E¯1∈ψ¯D1=x1,x2x1,y2|x1∈φ¯A1,x2y2∈ψ¯D22∪x1,x2y1,x2|x1y1∈ψ¯D12,x2∈φ¯A2⇒ψ¯D1□ψ¯D22=ψ¯D12□ψ¯D22.
As the vertex setℛ¯1□ℛ¯22 and ℛ¯12□ℛ¯22 is also same, so ℛ¯1□ℛ¯22=ℛ¯12□ℛ¯22. Similarly, ℛ¯1□ℛ¯22=ℛ¯12□ℛ¯22. Hence, ℛ1□ℛ22=ℛ12□ℛ22.Definition 17.
Letℛ1L2=φ¯A1,ψ¯D12,L¯1,φ¯A1,ψ¯D12,L¯1 and ℛ2L2 = φ¯A2,ψ¯D22,L¯2,φ¯A2,ψ¯D22,L¯2 be L2-sections of two hypergraphs ℛ1 and ℛ2. The Cartesian product of L2-sections is a rough hypergraph with a labeling function L1□L2=L¯1□L¯2,L¯1□L¯2, where L¯1□L¯2:ψ¯D12×ψ¯D22⟶ψ¯D1□ψ¯D2 and L¯1□L¯2:ψ¯D12×ψ¯D22⟶ψ¯D1□ψ¯D2 are defined as(10)L¯1□L¯2k1×l2,g2=k1×E¯2|E¯2∈L¯2l2g2L¯1□L¯2l1,g1×k2=E¯1×k2|E¯1∈L¯1l1g1L¯1□L¯2k1×l2,g2=k1×E¯2|E¯2∈L¯2l2g2L¯1□L¯2l1,g1×k2=E¯1×k2|E¯1∈L¯1l1g1.Lemma 6.
Letℛ1 and ℛ2 be two rough hypergraphs, then(1)
ℛ1□ℛ2L2=ℛ1L2□ℛ2L2(2)
ℛ1□ℛ2L2−1=ℛ1L2−1□ℛ2L2−1Proof.
(1)
It is clear from Lemma7 that ℛ1□ℛ22=ℛ12□ℛ22. It only needs to prove that the labeling function of ℛ1□ℛ2L2 and the function L1□L2 of Definition 17 are the same. Let ℛ1□ℛ2L2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D22,L¯,φ¯A1×φ¯A2,ψ¯D1□ψ¯D22,L¯ be the L2-section of ℛ1□ℛ2 with a labeling function L=L¯,L¯, where L¯:ψ¯D1□ψ¯D22⟶ψ¯D1□ψ¯D2 and L¯:ψ¯D1□ψ¯D22⟶ψ¯D1□ψ¯D2 are mappings. It is to be shown that L¯k1,k2g1,g2=L¯1□L¯2k1,k2g1,g2 and L¯k1,k2g1,g2=L¯1□L¯2k1,k2g1,g2. By Definitions 11 and 17,(11)L¯k1,k2k1,g2=E¯|k1,k2,k1,g2⊆E¯∈ψ¯D1□ψ¯D2=k1×E¯2|k1,k2,k1,g2⊆k1×E¯2∈ψ¯D1□ψ¯D2=k1×E¯2|k2,g2⊆E¯2∈ψ¯D2=k1×E¯2|E¯2∈L¯2k2g2⇒L¯k1,k2k1,g2=L¯1□L¯2k1,k2k1,g2.⇒L¯k1,k2g1,k2=E¯|k1,k2,g1,k2⊆E¯∈ψ¯D1□ψ¯D2=E¯1×k2|k1,k2,g1,k2⊆E¯1×k2∈ψ¯D1□ψ¯D2=E¯1×k2|k1,g1⊆E¯1∈ψ¯D1=E¯1×k2|E¯1∈L¯1k1g1⇒L¯k1,k2g1,k2=L¯1□L¯2k1,k2g1,k2.Thus,L¯k1,k2g1,g2=L¯1□L¯2k1,k2g1,g2, for each k1,k2g1,g2∈ψ¯D1□ψ¯D22. Similarly, L¯=L¯1□L¯2. Hence, L=L1□L2 which clearly proves ℛ1□ℛ2L2=ℛ1L2□ℛ2L2.(2)
For any rough hypergraphℛ, ℛL2−1=ℛ,L is the rough hypergraph with a labeling function. The proof of part 2 is clear from part 1 and Remark 1. Using Remark 1, ℛ1□ℛ2L2−1=ℛ1□ℛ2,L. By proof of part 1, ℛ1□ℛ2,L=ℛ1□ℛ2,L1□L2=ℛ1,L1□ℛ2,L2=ℛ1L2−1□ℛ2L2−1. □Theorem 6.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2; then, for any k1,g1∈Q1 and k2,g2∈Q2,(12)dℛ1□ℛ2k1,k2,g1,g2=dℛ1k1,g1+dℛ2k2,g2.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 ℛ1 and ℛ2,(13)dℛ1□ℛ2k1,k2,g1,g2=dℛ1□ℛ22k1,k2,g1,g2=dℛ12□ℛ22k1,k2,g1,g2=dℛ12k1,g1+dℛ22k2,g2⇒dℛ1□ℛ2k1,k2,g1,g2=dℛ1k1,g1+dℛ2k2,g2.
Letℛ be a rough hypergraph. If there exist prime rough hypergraphs ℛ1,ℛ2,…,ℛq such that ℛ≅ℛ1□ℛ2□…□ℛq, then it is called PFD (prime factor decomposition) of ℛ into q 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.Algorithm 1: Method to compute PFD of a rough hypergraph.
(1)
Given a connected rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D(2)
Compute the L2-sectionℛ2=φ¯A,ψ¯D2,φ¯A,ψ¯D2 of ℛ(3)
Using the Algorithm of Imrich and Peterin [52], decompose the labeled graphs φ¯A,ψ¯D2 and φ¯A,ψ¯D2 into prime factors w.r.t the Cartesian product. That is, the edges of φ¯A,ψ¯D2 and φ¯A,ψ¯D2 are colored w.r.t the copies of corresponding prime factors.(4)
Merge the factors if necessary(5)
Computeℛ2−1 using the labeled and the PFD of ℛ is obtained such that the colored copies are the prime factorsDefinition 18.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2, respectively. A homomorphism from ℛ1 into ℛ2 is a mapping f:Q1⟶Q2 if there exist homomorphisms f¯:Q1⟶Q2 and f¯:Q1⟶Q2; that is,(1)
Iff¯D is a hyperedge in ℛ2¯, then D is a hyperedge in ℛ1¯(2)
Iff¯D is a hyperedge in ℛ2¯, then D is a hyperedge in ℛ1¯
A homomorphism which is a one-to-one correspondence betweenQ1 and Q2 is called an isomorphism. In this case, we say that the rough hypergraphs ℛ1 and ℛ2 are isomorphic to each other and write as ℛ1≅ℛ2.Definition 19.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D and ℛ′=φ¯A′,ψ¯D′,φ¯A′,ψ¯D′ be two rough hypergraphs on Q and Q′, respectively. A surjective homomorphism f:Q′⟶Q is called a covering projection if(1)
f¯−1k=f¯−1E¯=p, for all k∈φ¯A, E∈ψ¯D(2)
f¯−1k=f¯−1E¯=p, for all k∈φ¯A, E∈ψ¯D(3)
E¯′∩F¯′=∅, for all distinct E¯′,F¯′∈f−1E¯, E¯∈ψ¯D(4)
E¯′∩F¯′=∅, for all distinct E¯′,F¯′∈f−1E¯, E¯∈ψ¯D
The rough hypergraphℛ′ is called a p-fold covering of ℛ, and ℛ is called quotient rough hypergraph of ℛ′. If p=2, ℛ′ is called a double cover of ℛ.Definition 20.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The sets S¯⊆φ¯A and S¯⊆φ¯A are called independent if they contain no hyperedge of ℛ¯ and ℛ¯, respectively. The cardinalities of the largest independent sets are denoted by βℛ¯ and βℛ¯ and are called the independence number of ℛ¯ and ℛ¯, respectively. The value βℛ=1/2βℛ¯+βℛ¯ is called independence number of ℛ.Definition 21.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The subsets T¯⊆φ¯A and T¯⊆φ¯A are called covers of ℛ¯ and ℛ¯, respectively, if T¯∩E¯≠∅ and T¯∩E¯≠∅, for each E¯∈ψ¯D and E¯∈ψ¯D. The cardinalities of minimal covers are denoted by τℛ¯ and τℛ¯ and are called covering numbers of ℛ¯ and ℛ¯, respectively. The average value τℛ=1/2τℛ¯+τℛ¯ is called covering number of ℛ.Definition 22.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The fractional covers of ℛ¯ and ℛ¯ are, respectively, the mappings f¯:φ¯A⟶ℝ+∪0 and f¯:φ¯A⟶ℝ+∪0 such that(14)∑k∈E¯f¯k≥1,for eachE¯∈ψ¯D,∑k∈E¯f¯k≥1,for eachE¯∈ψ¯D.
The valueτ∗ℛ=1/2minf¯∑k∈φ¯Af¯k+minf¯∑k∈φ¯Af¯k is called fractional covering number (FC number) of ℛ.Definition 23.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The subsets M¯⊆ψ¯D and M¯⊆ψ¯D are called matching if every pair of hyperedges from M¯ and M¯ are mutually disjoint. The cardinalities of maximal matchings are denoted by νℛ¯ and νℛ¯ and are called matching numbers of ℛ¯ and ℛ¯, respectively. The matching number of ℛ is computed as νℛ=1/2νℛ¯+νℛ¯Definition 24.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The minimum number of mutually disjoint hyperedges whose union is the sets of vertices φ¯A and φ¯A is called the partition number of ℛ¯ and ℛ¯, respectively, denoted by ρℛ¯ and ρℛ¯. The value ρℛ=1/2ρℛ¯+ρℛ¯ is called the partition number of ℛ. If such partitions do not exist, then ρℛ=∞.
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. Letℛ1⊛ℛ2 denote any product of two rough hypergraphs ℛ1 and ℛ2. For any rough hypergraph ℛ, if there exists another rough hypergraph U such that ℛ⊛U=ℛ, then U is called the unit element. Note that U must be a hypergraph with a single vertex and no loops.Definition 25.
A rough hypergraphℛ=ℛ¯,ℛ¯ is called conformal if ℛ¯ and ℛ¯ are both conformal hypergraphs. That is, corresponding to each clique of 2-section ℛ¯2 (and ℛ¯2), there is a hyperedge in ℛ¯ (and ℛ¯).Definition 26.
Letℛ=ℛ¯,ℛ¯ be a rough hypergraph. The collection of all hyperedges in ℛ¯ (and ℛ¯) containing a common vertex k is called a star of ℛ¯ (and ℛ¯), denoted by ℛ¯k (and ℛ¯k). The pair ℛk=ℛ¯k,ℛ¯k is called a rough star of ℛ. The subsets E¯⊆ψ¯D and E¯⊆ψ¯D are called intersecting families of ℛ¯ and ℛ¯ if every pair of hyperedges of E¯ and E¯ have nonempty intersection. The pair E¯,E¯ is called a rough intersecting family of ℛ. A rough hypergraph ℛ is said to satisfy Helly property if each rough intersecting family in ℛ is a rough star.Definition 27.
Letℛ1′ and ℛ2′ be p1-fold and p2-fold coverings of rough hypergraphs ℛ1 and ℛ2 via covering projections f1=f¯1,f¯1 and f2=f¯2,f¯2, respectively; then, the Cartesian product f¯,f¯=f=f1□f2=f¯1□f¯2,f¯1□f¯2 is defined as(1)
f¯k1,k2=f¯1k1,f¯2k2, for all k1,k2∈φ¯A1×φ¯A2(2)
f¯k1,k2=f¯1k1,f¯2k2, for all k1,k2∈φ¯A1×φ¯A2(3)
f¯k1×E¯2=f¯1k1×f¯2E¯2, for all k1∈φ¯A1, E¯2∈ϕ¯D2(4)
f¯k1×E¯2=f¯1k1×f¯2E¯2, for all k1∈φ¯A1, E¯2∈ϕ¯D2(5)
f¯E¯1×k2=f¯1E¯1×f¯2k2, for all E¯1∈ϕ¯D1, k1∈φ¯A1(6)
f¯E¯1×k2=f¯1E¯1×f¯2k2, for all E¯1∈ϕ¯D1, k1∈φ¯A1Theorem 7.
Letℛ′1 and ℛ′2 be p1-fold and p2-fold coverings of rough hypergraphs ℛ1 and ℛ2 via covering projections f1 and f2, respectively; then, ℛ′1□ℛ′2 is a p1p2-fold covering of ℛ1□ℛ2 via covering projection f1□f2.Proof.
The mappingf=f1□f2 is given in Definition 27. We first need to show that f=f¯,f¯ is a surjective homomorphism. Let k1′×E¯′2 be a hyperedge in ℛ′1□ℛ′2; then, f¯k1′×E¯′2=f¯1k1′×f¯2E¯′2. Since f¯1 and f¯2 are homomorphisms, therefore f¯1k1′∈φ¯A2′ and f¯2E¯′2 is a hyperedge in ℛ¯2. Thus, f¯1k1′×f¯2E¯′2 is a hyperedge in ℛ¯1□ℛ¯2. Similarly, f¯E¯′1×k2′ is a hyperedge ℛ¯1□ℛ¯2 and f¯k1′×E¯′2; f¯E¯′1×k2′ are hyperedges in ℛ¯1□ℛ¯2 showing that f is a homomorphism. The surjectivity of f¯1□f¯2 and f¯1□f¯2 is obvious from the surjectivity of f¯1, f¯2, f¯1, and f¯2.
Letk1,k2∈φ¯A1×φ¯A2 be a vertex in ℛ¯1□ℛ¯2; then,(15)f¯1−1k1,k2=k1′,k2′∣f¯1k1′=k1,f¯2k2′=k2⇒f¯1−1k1,k2=f¯1−1k1×f¯2−1k2⇒f¯1−1k1,k2=f¯1−1k1×f¯2−1k2=p1p2.
Similarly,(16)f¯−1k1,k2=f¯1−1k1×f¯2−1k2=p1p2.
Consider a hyperedgek1×E¯2 in ℛ¯1□ℛ¯2; then,(17)f¯−1k1×E¯2=k1′,k2′|f¯1k1′=k1,f¯2k2′∈E¯2=f¯1−1k1×f¯2−1k2|f¯2k2′∈E¯2=f¯1−1k1×f¯2−1E¯2⇒f¯−1k1×E¯2=f¯1−1k1×f¯2−1E¯2=p1p2.
Similarly,(18)f¯1−1E¯1×k2=f¯1−1E¯1×f¯2−1k2=p1p2f¯−1k1×E¯2=f¯−1E¯1×k2=p1p2.
Letk1×E¯2 be a hyperedge in ℛ¯1□ℛ¯2 such that k1′×E¯′2,k1′×F¯′2∈f¯1−1k1×E¯2 and k1′×E¯′2∩k1′×F¯′2≠∅. It follows that E¯′2∩F¯′2≠∅, a contradiction. Similarly, we can prove the other cases. Hence, f=f¯,f¯ is covering projection and ℛ′1□ℛ′2 is a p1p2-fold covering of ℛ1□ℛ2. □Theorem 8.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1□ℛ2 is conformal if and only if ℛ1 and ℛ2 are conformal.Proof.
Letℛ1 and ℛ2 be two conformal rough hypergraphs. Let ℒ=ℒ¯,ℒ¯ be a rough clique in ℛ1□ℛ22; then, by Lemma 7, there are two possibilities.Case 1.
There exists a rough cliqueℒ1=ℒ¯1,ℒ¯1 in ℛ12 such that ℒ¯=ℒ¯1×x¯2, for some x¯2∈ψ¯D2, and ℒ¯=ℒ¯1×x¯2, for some x¯2∈ψ¯D2. Since ℒ is a rough clique, therefore, ℒ1 is also a rough clique. As ℛ1 is a conformal rough hypergraph, therefore there exist hyperedges E¯1 and E¯1 corresponding to ℒ¯1 and ℒ¯1 in ℛ¯1 and ℛ¯1. Thus, E¯1×x¯2,E¯1×x¯2 is a rough hyperedge corresponding to ℒ1 showing that ℛ1□ℛ2 is a conformal rough hypergraph.Case 2.
There exists a rough cliqueℒ2=ℒ¯2,ℒ¯2 in ℛ22 such that ℒ¯=x¯1×ℒ¯2, for some x¯1∈ψ¯D1, and ℒ¯=x¯1×ℒ¯2, for some x¯1∈ψ¯D1. This case can be proved along the same lines as Case 1. Hence, ℛ1□ℛ2 is a conformal rough hypergraph.
Conversely, letℛ1□ℛ2 be a conformal rough hypergraph. Let ℒ1 and ℒ2 be rough cliques in ℛ1 and ℛ2, respectively. By Lemma 7, ℒ=ℒ1□ℒ2 is a rough clique in ℛ1□ℛ22. Since ℛ1□ℛ2 is a conformal rough hypergraph, therefore, there exists a rough hyperedge E=E¯,E¯ in ℛ1□ℛ2 corresponding to ℒ. By Lemma 7, there exist rough hyperedges E1 and E2 in ℛ1 and ℛ2 such that E=E1□E2. Clearly, E1 and E2 correspond to in ℒ1 and ℒ2, respectively, proving that ℛ1□ℛ2 are conformal rough hypergraphs. □Theorem 9.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1□ℛ2 has the Helly property iff ℛ1 and ℛ2 have the Helly property.Definition 28.
The dual of a rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is a rough hypergraph ℛ∗=φ¯A∗,ψ¯D∗,φ¯A∗,ψ¯D∗, where(1)
The hyperedge setψ¯D of ℛ is the vertex set of ℛ∗, that is, φ¯A∗=ψ¯D=E¯1,E¯2,…,E¯r(2)
The edge setψ¯D of ℛ is the vertex set of ℛ∗, that is, φ¯A∗=ψ¯D=E¯1,E¯2,…,E¯r(3)
If|φ¯A|=m, then ψ¯D∗=K¯1,K¯2,…,K¯m is the hyperedge set of ℛ∗ such that K¯i=E¯j|ki∈E¯j, that is, K¯i is the collection of those hyperedges of ℛ¯ which share the common vertex ki(4)
Ifφ¯A=n, then ψ¯D∗=K¯1,K¯2,…,K¯n is the hyperedge set of ℛ∗ such that K¯i=E¯j|ki∈E¯j, that is, K¯i is the collection of those hyperedges of ℛ¯ which share the common vertices kiRemark 3.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1□ℛ2∗ may not be equal to ℛ1∗□ℛ2∗. Since, for any two hypergraphs H1 and H2, H1□H2∗ is not equal to h1∗□H2∗ 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 thatH1□H2 has nine edges and so H1□H2∗ has nine vertices. But, H1∗□H2∗ has three vertices. The vertex sets of H1□H2∗ and H1∗□H2∗ are not equal and it proves our claim.Figure 7
HypergraphH1.Figure 8
HypergraphH2.


### 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.
Letℛ1∗ℛ2∗⋯∗ℛr be any product of rough hypergraphs and Vℛi denotes the vertex set of ℛi, for any 1≤i≤r. The mapping pi=p¯i,p¯i:Vℛ1∗ℛ2∗⋯∗ℛr⟶Vℛi is called the projection of ℛ1∗ℛ2∗⋯∗ℛr onto ith factor ℛi, where p¯i:ℛ¯1∗ℛ¯2∗⋯∗ℛ¯r⟶ℛ¯i and p¯i:ℛ¯1∗ℛ¯2∗⋯∗ℛ¯r⟶ℛ¯i are the projection mappings defined as(19)p¯ik1,k2,…,kn=p¯ik1,k2,…,kn=ki,for allki∈Qi,1≤i≤r.Definition 30.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The square product of ℛ1 and ℛ2 is a rough hypergraph ℛ1×ℛ2=ℛ¯1×ℛ¯2,ℛ¯1×ℛ¯2 which is defined as(1)
ℛ¯1×ℛ¯2=φ¯A1×φ¯A2,ψ¯D1×ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×ψ¯D2=E¯1×E¯2|E¯1∈ψ¯D1,E¯2∈ψ¯D2(2)
ℛ¯1□ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×ψ¯D2=E¯1×E¯2|E¯1∈ψ¯D1,E¯2∈ψ¯D2
In short,ℛ¯1×ℛ¯2 and ℛ¯1×ℛ¯2 are the square products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1×ℛ2×ℛ3≅ℛ1×ℛ2×ℛ3(2)
ℛ1×ℛ2≅ℛ2×ℛ1(3)
ℛ1×ℛ2∪ℛ3≅ℛ1×ℛ2∪ℛ1×ℛ2(4)
ℛ×U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1×ℛ2⟶ℛ1 and p2:Vℛ1×ℛ2⟶Vℛ2 are homomorphismsDefinition 31.
Letℛ be a rough hypergraph; then, ℛ is called an r-uniform rough hypergraph if ℛ¯ and ℛ¯ are both r-uniform hypergraphs; that is, for each E¯∈ψ¯D and E¯∈ψ¯D, E¯=E¯=r.Lemma 7.
Letℛ1 and ℛ2 be two r-uniform rough hypergraphs; then, ℛ1×ℛ22≅ℛ12×ℛ22.Proof.
Since the vertex set ofℛi and ℛi2 is the same for i=1,2, therefore, the vertex set of ℛ1×ℛ22 and ℛ12×ℛ22 is the same. It only needs to show that the set of hyperedges of ℛ1×ℛ22 and ℛ12×ℛ22 is the same. By Definition 30, ℛ1×ℛ22=ℛ¯1×ℛ¯22,ℛ¯1×ℛ¯22, where ℛ¯1×ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×ψ¯D22 and ℛ¯1×ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×ψ¯D22. As ℛ1 and ℛ2 are r-uniform rough hypergraphs, so we have(20)ψ¯D1×ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1×ψ¯D2=x1,x2x1,y2|x1,y1⊆E¯1∈ψ¯D1,x2,y2⊆E¯2∈ψ¯D2,E¯1=E¯2=x1,x2x1,y2|x1y1∈ψ¯D12,x2y2∈ψ¯D22⇒ψ¯D1×ψ¯D22=ψ¯D12×ψ¯D22.
As the vertex setℛ¯1×ℛ¯22 and ℛ¯12×ℛ¯22 is also the same, so ℛ¯1×ℛ¯22≅ℛ¯12×ℛ¯22. Similarly, ℛ¯1×ℛ¯22≅ℛ¯12×ℛ¯22. Hence, ℛ1×ℛ22≅ℛ12×ℛ22.Theorem 10.
Letℛ1 and ℛ2 be two rough hypergraphs; then,(1)
rℛ1×ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(2)
sℛ1×ℛ2=maxsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Proof.
Since, by Definition30, ℛ1×ℛ2=ℛ¯1×ℛ¯2,ℛ¯1×ℛ¯2, we first need to compute rℛ¯1×ℛ¯2 and rℛ¯1×ℛ¯2. By Definition 6,(21)rℛ¯1×ℛ¯2=maxE¯∈ψ¯D1×ψ¯D2E¯=maxE¯|p1E¯=E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=maxmaxE¯1∈ψ¯D1E¯1,maxE¯2∈ψ¯D2E¯2⇒rℛ¯1×ℛ¯2=maxrℛ¯1,rℛ¯2.
Similarly,rℛ¯1×ℛ¯2=maxrℛ¯1,rℛ¯2. Hence,(22)rℛ1×ℛ2=12maxrℛ¯1,rℛ¯2+maxrℛ¯1,rℛ¯2=maxrℛ¯1+rℛ¯12,rℛ¯2+rℛ¯22,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12⇒rℛ¯1×ℛ¯2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12.
Using Definition6, the antirank of ℛ1×ℛ2 is given as(23)sℛ¯1×ℛ¯2=minE¯∈ψ¯D1×ψ¯D2E¯=minE¯|p1E¯=E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=maxminE¯1∈ψ¯D1E¯1,minE¯2∈ψ¯D2E¯2⇒sℛ¯1×ℛ¯2=maxsℛ¯1,sℛ¯2.
Similarly,sℛ¯1×ℛ¯2=maxsℛ¯1,sℛ¯2. Hence,(24)sℛ1×ℛ2=12maxsℛ¯1,sℛ¯2+maxsℛ¯1,sℛ¯2=maxsℛ¯1+sℛ¯12,sℛ¯2+sℛ¯22,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12⇒sℛ1×ℛ2=maxsℛ1,sℛ2,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12.


### 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The direct product ×⌣ of ℛ1 and ℛ2 is a rough hypergraph ℛ1×⌣ℛ2=ℛ¯1×⌣ℛ¯2,ℛ¯1×⌣ℛ¯2 which is defined as(1)
ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×⌣ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯∈ψ¯D2(2)
ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×⌣ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯∈ψ¯D2
In short,ℛ¯1×⌣ℛ¯2 and ℛ¯1×⌣ℛ¯2 are the MRP direct products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1×⌣ℛ2×⌣ℛ3≅ℛ1×⌣ℛ2×⌣ℛ3(2)
ℛ1×⌣ℛ2≅ℛ2×⌣ℛ1(3)
ℛ1×⌣ℛ2∪ℛ3≅ℛ1×⌣ℛ2∪ℛ1×⌣ℛ2(4)
ℛ×⌣U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1×⌣ℛ2⟶ℛ1 and p2:Vℛ1×⌣ℛ2⟶Vℛ2 may not be weak homomorphismsLemma 8.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1×⌣ℛ22≅ℛ12□ℛ22∪ℛ12×ℛ22.Proof.
Since the vertex set ofℛi and ℛi2 is the same for i=1,2, therefore, the vertex set of ℛ1×⌣ℛ22 and the union of ℛ12□ℛ22 and ℛ12×ℛ22 is the same. It only needs to show that the set of hyperedges of ℛ1×⌣ℛ22 and ℛ12□ℛ22∪ℛ12×ℛ22 is the same. By Definition 32, ℛ1×⌣ℛ22=ℛ¯1×⌣ℛ¯22,ℛ¯1×⌣ℛ¯22, where ℛ¯1×⌣ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D22 and ℛ¯1×⌣ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D22. So, we have(25)ψ¯D1×⌣ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1×ψ¯D2ψ¯D1×⌣ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1×ψ¯D2=x1,x2y1,y2|x1=y1,x2,y2⊆p2E¯∈ψ¯D2∪x1,x2x1,y2|x1,y1⊆p1E¯∈ψ¯D1,x2=y2∪x1,x2x1,y2∣x1,y1⊆p1E¯∈ψ¯D1,x2,y2⊆p2E¯∈ψ¯D2,E¯1=E¯2=x1,x2x1,y2|x1,y1∈φ¯A1,x2y2∈ψ¯D22x1,x2x1,y2|x1y1∈ψ¯D12,x2,y2∈φ¯A2x1,x2x1,y2|x1y1∈ψ¯D12,x2y2∈ψ¯D22⇒ψ¯D1×⌣ψ¯D22=ψ¯D12□ψ¯D22∪ψ¯D12×ψ¯D22.
Thus,ℛ¯1×⌣ℛ¯22≅ℛ¯12□ℛ¯22∪ℛ¯12×ℛ¯22. Similarly, ℛ¯1×⌣ℛ¯22≅ℛ¯12□ℛ¯22∪ℛ¯12×ℛ¯22. Hence, ℛ1×ℛ22≅ℛ12□ℛ22∪ℛ12×ℛ22.Theorem 11.
Letℛ1 and ℛ2 be two rough hypergraphs; then,(1)
rℛ1×⌣ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(2)
sℛ1×⌣ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Proof.
Since, by Definition32, ℛ1×⌣ℛ2=ℛ¯1×⌣ℛ¯2,ℛ¯1×⌣ℛ¯2, we first need to compute rℛ¯1×⌣ℛ¯2 and rℛ¯1×⌣ℛ¯2. By Definition 6,(26)rℛ¯1×⌣ℛ¯2=maxE¯∈ψ¯D1×⌣ψ¯D2E¯=maxE¯|p1E¯=E¯1∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=maxmaxE¯1∈ψ¯D1E¯1,maxE¯2∈ψ¯D2E¯2⇒rℛ¯1×⌣ℛ¯2=maxrℛ¯1,rℛ¯2.
Similarly,rℛ¯1×⌣ℛ¯2=maxrℛ¯1,rℛ¯2. Hence,(27)rℛ1×⌣ℛ2=12maxrℛ¯1,rℛ¯2+maxrℛ¯1,rℛ¯2=maxrℛ¯1+rℛ¯12,rℛ¯2+rℛ¯22,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12⇒rℛ1×⌣ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12.
Using Definition6, the antirank of ℛ1×⌣ℛ2 is given as(28)sℛ¯1×⌣ℛ¯2=minE¯∈ψ¯D1×⌣ψ¯D2E¯=minE¯|p1E¯=E¯1∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=minminE¯1∈ψ¯D1E¯1,minE¯2∈ψ¯D2E¯2⇒sℛ¯1×⌣ℛ¯2=minsℛ¯1,sℛ¯2.
Similarly,sℛ¯1×⌣ℛ¯2=maxsℛ¯1,sℛ¯2. Hence,(29)sℛ1×⌣ℛ2=12minsℛ¯1,sℛ¯2+minsℛ¯1,sℛ¯2=minsℛ¯1+sℛ¯12,sℛ¯2+sℛ¯22,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12⇒sℛ1×⌣ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12.


### 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The union f ℛ1 and ℛ2 is a rough hypergraph ℛ1∪ℛ2=ℛ¯1∪ℛ¯2,ℛ¯1∪ℛ¯2, where ℛ¯1∪ℛ¯2=φ¯A1∪φ¯A2,ψ¯D1∪ψ¯D2 and ℛ¯1∪ℛ¯2=φ¯A1∪φ¯A2,ψ¯D1∪ψ¯D2.
The intersectionfℛ1 and ℛ2 is a rough hypergraph ℛ1∩ℛ2=ℛ¯1∩ℛ¯2,ℛ¯1∩ℛ¯2, where ℛ¯1∩ℛ¯2=φ¯A1∩φ¯A2,ψ¯D1∩ψ¯D2 and ℛ¯1∩ℛ¯2=φ¯A1∩φ¯A2,ψ¯D1∩ψ¯D2.
In short,ℛ¯1∪ℛ¯2 and ℛ¯1∩ℛ¯2 are the union and intersection of lower approximate hypergraphs ℛ¯1 and ℛ¯2. 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 ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1∪ℛ2∪ℛ3≅ℛ1∪ℛ2∪ℛ3(2)
ℛ1∩ℛ2∩ℛ3≅ℛ1∩ℛ2∩ℛ3(3)
ℛ1∪ℛ2≅ℛ2∪ℛ1(4)
ℛ1∩ℛ2≅ℛ2∩ℛ1(5)
ℛ1∪ℛ2∩ℛ3≅ℛ1∪ℛ2∩ℛ1∪ℛ2(6)
The projectionsp1:Vℛ1∪ℛ2⟶ℛ1 and p2:Vℛ1∪ℛ2⟶Vℛ2 may not be weak homomorphisms(7)
The projectionsp1:Vℛ1∩ℛ2⟶ℛ1 and p2:Vℛ1∩ℛ2⟶Vℛ2 may not be weak homomorphismsRemark 4.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs, then(1)
ℛ1∪ℛ22≅ℛ12∪ℛ22(2)
rℛ1∪ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(3)
sℛ1∪ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2(4)
ℛ1∪ℛ2∗≅ℛ1∗∪ℛ2∗


### 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The strong product of ℛ1 and ℛ2 is a rough hypergraph ℛ1⊠ℛ2=ℛ¯1⊠ℛ¯2,ℛ¯1⊠ℛ¯2, where ℛ¯1⊠ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×ψ¯D2 and ℛ¯1∪ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×ψ¯D2. In other words, the strong product of ℛ1 and ℛ2 is the union of Cartesian product and square product of rough hypergraphs ℛ1 and ℛ2.
In short,ℛ¯1⊠ℛ¯2 and ℛ¯1⊠ℛ¯2 are the strong products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1⊠ℛ2⊠ℛ3≅ℛ1⊠ℛ2⊠ℛ3(2)
ℛ1⊠ℛ2≅ℛ2⊠ℛ1(3)
ℛ1⊠ℛ2∪ℛ3≅ℛ1⊠ℛ2∪ℛ1⊠ℛ2(4)
ℛ⊠U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1⊠ℛ2⟶ℛ1 and p2:Vℛ1⊠ℛ2⟶Vℛ2 may not be weak homomorphisms(6)
ℛ1⊠ℛ22≅ℛ12⊠ℛ22(7)
rℛ1⊠ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(8)
sℛ1⊠ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Theorem 12.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2; then, for any k1,g1∈Q1 and k2,g2∈Q2,(30)dℛ1⊠ℛ2k1,k2,g1,g2=dℛ1k1,g1,dℛ2k2,g2,dℛ¯1k1,g1+dℛ¯2k2,g22,dℛ¯1k1,g1+dℛ¯2k2,g22.Proof.
The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2, and the result ℛ1⊠ℛ22≅ℛ12⊠ℛ22. Thus, for any two rough hypergraphs ℛ1 and ℛ2,(31)dℛ¯1⊠ℛ¯2k1,k2,g1,g2=dℛ¯1⊠ℛ¯22k1,k2,g1,g2=dℛ¯12⊠ℛ¯22k1,k2,g1,g2=maxdℛ¯12k1,g1,dℛ¯22k2,g2⇒dℛ¯1⊠ℛ¯2k1,k2,g1,g2=maxdℛ¯1k1,g1,dℛ¯2k2,g2.
Similarly,dℛ¯1⊠ℛ¯2k1,k2,g1,g2=maxdℛ¯1k1,g1,dℛ¯2k2,g2, 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The strong product of ℛ1 and ℛ2 is a rough hypergraph ℛ1⊠⌣ℛ2=ℛ¯1⊠⌣ℛ¯2,ℛ¯1⊠⌣ℛ¯2, where ℛ¯1⊠⌣ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×⌣ψ¯D2 and ℛ¯1⊠⌣ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×⌣ψ¯D2. In other words, the strong product of ℛ1 and ℛ2 is the union of Cartesian product and direct product of rough hypergraphs ℛ1 and ℛ2.
In short,ℛ¯1⊠⌣ℛ¯2 and ℛ¯1⊠⌣ℛ¯2 are the strong products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1⊠⌣ℛ2⊠⌣ℛ3≅ℛ1⊠⌣ℛ2⊠⌣ℛ3(2)
ℛ1⊠⌣ℛ2≅ℛ2⊠⌣ℛ1(3)
ℛ1⊠⌣ℛ2∪ℛ3≅ℛ1⊠⌣ℛ2∪ℛ1⊠⌣ℛ2(4)
ℛ⊠⌣U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1⊠⌣ℛ2⟶ℛ1 and p2:Vℛ1⊠⌣ℛ2⟶Vℛ2 may not be weak homomorphisms(6)
ℛ1⊠⌣ℛ22≅ℛ12⊠ℛ22(7)
rℛ1⊠⌣ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(8)
sℛ1⊠⌣ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Theorem 13.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2, then for any k1,g1∈Q1 and k2,g2∈Q2,(32)dℛ1⊠⌣ℛ2k1,k2,g1,g2=dℛ1k1,g1,dℛ2k2,g2,dℛ¯1k1,g1+dℛ¯2k2,g22,dℛ¯1k1,g1+dℛ¯2k2,g22.Proof.
The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2 and the result ℛ1⊠⌣ℛ22≅ℛ12⊠ℛ22. Thus, for any two rough hypergraphs ℛ1 and ℛ2. □


### 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The lexicographic product ° of ℛ1 and ℛ2 is a rough hypergraph ℛ1°ℛ2=ℛ¯1°ℛ¯2,ℛ¯1°ℛ¯2 which is defined as(1)
ℛ¯1°ℛ¯2=φ¯A1×φ¯A2,ψ¯D1°ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1°ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪k1×E¯2|k1∈ψ¯D1,E¯2∈ψ¯D2(2)
ℛ¯1°ℛ¯2=φ¯A1×φ¯A2,ψ¯D1°ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1°ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪k1×E¯2|k1∈ψ¯D1,E¯2∈ψ¯D2
In short,ℛ¯1°ℛ¯2 and ℛ¯1°ℛ¯2 are the lexicographic products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U (left identity) as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1°ℛ2°ℛ3≅ℛ1°ℛ2°ℛ3(2)
ℛ1°ℛ2≅ℛ2°ℛ1(3)
ℛ1°ℛ2∪ℛ3≅ℛ1°ℛ2∪ℛ1°ℛ2(4)
U°ℛ≅ℛ, where U is a single vertex hypergraph without loops(5)
ℛ°U≅ℛ(6)
The projectionsp1:Vℛ1°ℛ2⟶ℛ1 and p2:Vℛ1°ℛ2⟶Vℛ2 may not be weak homomorphisms(7)
ℛ1°ℛ22≅ℛ12°ℛ22(8)
rℛ1°ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(9)
sℛ1°ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Theorem 14.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2; then, for any k1,g1∈Q1 and k2,g2∈Q2,(33)dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2,ifk1=g1,degk1≠0,dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2,ifk1=g1,degk1≠0.Proof.
The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2, and the result ℛ1⊠ℛ22≅ℛ12⊠ℛ22. Thus, for any two rough hypergraphs ℛ1 and ℛ2,(34)dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1°ℛ¯22k1,k2,g1,g2=dℛ¯12°ℛ¯22k1,k2,g1,g2=dℛ¯12k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2ifk1=g1,degk1≠0,⇒dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2,ifk1=g1,degk1≠0.
Similarly for upper approximate hypergraphs, the result follows.Definition 37.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The costrong product ⋆ of ℛ1 and ℛ2 is a rough hypergraph ℛ1⋆ℛ2=ℛ¯1⋆ℛ¯2,ℛ¯1⋆ℛ¯2 which is defined as ℛ1⋆ℛ2=ℛ1°ℛ2∪ℛ2°ℛ1.
In short,ℛ¯1°ℛ¯2 and ℛ¯1°ℛ¯2 are the costrong products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U (left identity) as a trivial hypergraph with a single vertex such that U¯=U¯.Remark 5.
Letℛ1 and ℛ2 be two rough hypergraphs, then(1)
ℛ1⋆ℛ22=ℛ12⋆ℛ22(2)
rℛ1⋆ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(3)
sℛ1⋆ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2


### 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.


## 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.
Letℛ1 =  ℛ¯1,ℛ¯1 and ℛ2 =  ℛ¯2,ℛ¯2 be two rough hypergraphs. The Cartesian product of ℛ1 and ℛ2 is a rough hypergraph ℛ1□ℛ2=ℛ¯1□ℛ¯2,ℛ¯1□ℛ¯2 which is defined as(1)
ℛ¯1□ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1□ψ¯D2=k1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2(2)
ℛ¯1□ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1□ψ¯D2=k1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2
In short,ℛ¯1□ℛ¯2 and ℛ¯1□ℛ¯2 are the Cartesian products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1□ℛ2□ℛ3≅ℛ1□ℛ2□ℛ3(2)
ℛ1□ℛ2≅ℛ2□ℛ1(3)
ℛ1□ℛ2∪ℛ3≅ℛ1□ℛ2∪ℛ1□ℛ2(4)
ℛ□U≅ℛ, where U is a single vertex rough hypergraph without loopsTheorem 5.
Letℛ1 and ℛ2 be two rough hypergraphs; then,(1)
rℛ1□ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(2)
sℛ1□ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Proof.
Since, by Definition16, ℛ1□ℛ2=ℛ¯1□ℛ¯2,ℛ¯1□ℛ¯2, we first need to compute rℛ¯1□ℛ¯2 and rℛ¯1□ℛ¯2. By Definition 6,(5)rℛ¯1□ℛ¯2=maxE∈ψ¯D1□ψ¯D2E=maxk1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2=maxE¯2|E¯2∈ψ¯D2∪E¯1|E¯1∈ψ¯D1=maxmaxE¯1∈ψ¯D1E¯1,maxE¯2∈ψ¯D2E¯2⇒rℛ¯1□ℛ¯2=maxrℛ¯1,rℛ¯2.
Similarly,rℛ¯1□ℛ¯2=maxrℛ¯1,rℛ¯2. Hence,(6)rℛ1□ℛ2=12maxrℛ¯1,rℛ¯2+maxℛ¯1,rℛ¯2=maxrℛ¯1+rℛ¯12,rℛ¯2+rℛ¯22,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12=maxrℛ1,rℛ2,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12.
By Definition6, the antirank of ℛ1□ℛ2 is given as(7)sℛ¯1□ℛ¯2=minE∈ψ¯D1□ψ¯D2E=mink1×E¯2|k1∈φ¯A1,E¯2∈ψ¯D2∪E¯1×k2|E¯1∈ψ¯D1,k2∈φ¯A2=minE¯2|E¯2∈ψ¯D2∪E¯1|E¯1∈ψ¯D1=minminE¯1∈ψ¯D1E¯1,minE¯2∈ψ¯D2E¯2⇒sℛ¯1□ℛ¯2=minsℛ¯1,sℛ¯2.
Similarly,sℛ¯1□ℛ¯2=minsℛ¯1,sℛ¯2. Hence,(8)sℛ1□ℛ2=12minsℛ¯1,sℛ¯2+minsℛ¯1,sℛ¯2=minsℛ¯1+sℛ¯12,sℛ¯2+sℛ¯22,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12=minsℛ1,sℛ2,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12.Lemma 5.
Letℛ1 and ℛ2 be two rough hypergraphs, then ℛ1□ℛ22=ℛ12□ℛ22.Proof.
Since the vertex set ofℛi and ℛi2 is the same for i=1,2, therefore the vertex set of ℛ1□ℛ22 and ℛ12□ℛ22 is the same. It only needs to show that the set of hyperedges of ℛ1□ℛ22 and ℛ12□ℛ22 is the same. By Definition 16, ℛ1□ℛ22=ℛ¯1□ℛ¯22,ℛ¯1□ℛ¯22, where ℛ¯1□ℛ¯22=φ¯A1×φ¯A2,ψ¯D1□ψ¯D22 and ℛ¯1□ℛ¯22=φ¯A1×φ¯A2,ψ¯D1□ψ¯D22. So, we have(9)ψ¯D1□ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1□ψ¯D2=x1,x2x1,y2|y1=x1,x2,y2⊆E¯2∈ψ¯D2∪x1,x2y1,x2|y2=x2,x1,y1⊆E¯1∈ψ¯D1=x1,x2x1,y2|x1∈φ¯A1,x2y2∈ψ¯D22∪x1,x2y1,x2|x1y1∈ψ¯D12,x2∈φ¯A2⇒ψ¯D1□ψ¯D22=ψ¯D12□ψ¯D22.
As the vertex setℛ¯1□ℛ¯22 and ℛ¯12□ℛ¯22 is also same, so ℛ¯1□ℛ¯22=ℛ¯12□ℛ¯22. Similarly, ℛ¯1□ℛ¯22=ℛ¯12□ℛ¯22. Hence, ℛ1□ℛ22=ℛ12□ℛ22.Definition 17.
Letℛ1L2=φ¯A1,ψ¯D12,L¯1,φ¯A1,ψ¯D12,L¯1 and ℛ2L2 = φ¯A2,ψ¯D22,L¯2,φ¯A2,ψ¯D22,L¯2 be L2-sections of two hypergraphs ℛ1 and ℛ2. The Cartesian product of L2-sections is a rough hypergraph with a labeling function L1□L2=L¯1□L¯2,L¯1□L¯2, where L¯1□L¯2:ψ¯D12×ψ¯D22⟶ψ¯D1□ψ¯D2 and L¯1□L¯2:ψ¯D12×ψ¯D22⟶ψ¯D1□ψ¯D2 are defined as(10)L¯1□L¯2k1×l2,g2=k1×E¯2|E¯2∈L¯2l2g2L¯1□L¯2l1,g1×k2=E¯1×k2|E¯1∈L¯1l1g1L¯1□L¯2k1×l2,g2=k1×E¯2|E¯2∈L¯2l2g2L¯1□L¯2l1,g1×k2=E¯1×k2|E¯1∈L¯1l1g1.Lemma 6.
Letℛ1 and ℛ2 be two rough hypergraphs, then(1)
ℛ1□ℛ2L2=ℛ1L2□ℛ2L2(2)
ℛ1□ℛ2L2−1=ℛ1L2−1□ℛ2L2−1Proof.
(1)
It is clear from Lemma7 that ℛ1□ℛ22=ℛ12□ℛ22. It only needs to prove that the labeling function of ℛ1□ℛ2L2 and the function L1□L2 of Definition 17 are the same. Let ℛ1□ℛ2L2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D22,L¯,φ¯A1×φ¯A2,ψ¯D1□ψ¯D22,L¯ be the L2-section of ℛ1□ℛ2 with a labeling function L=L¯,L¯, where L¯:ψ¯D1□ψ¯D22⟶ψ¯D1□ψ¯D2 and L¯:ψ¯D1□ψ¯D22⟶ψ¯D1□ψ¯D2 are mappings. It is to be shown that L¯k1,k2g1,g2=L¯1□L¯2k1,k2g1,g2 and L¯k1,k2g1,g2=L¯1□L¯2k1,k2g1,g2. By Definitions 11 and 17,(11)L¯k1,k2k1,g2=E¯|k1,k2,k1,g2⊆E¯∈ψ¯D1□ψ¯D2=k1×E¯2|k1,k2,k1,g2⊆k1×E¯2∈ψ¯D1□ψ¯D2=k1×E¯2|k2,g2⊆E¯2∈ψ¯D2=k1×E¯2|E¯2∈L¯2k2g2⇒L¯k1,k2k1,g2=L¯1□L¯2k1,k2k1,g2.⇒L¯k1,k2g1,k2=E¯|k1,k2,g1,k2⊆E¯∈ψ¯D1□ψ¯D2=E¯1×k2|k1,k2,g1,k2⊆E¯1×k2∈ψ¯D1□ψ¯D2=E¯1×k2|k1,g1⊆E¯1∈ψ¯D1=E¯1×k2|E¯1∈L¯1k1g1⇒L¯k1,k2g1,k2=L¯1□L¯2k1,k2g1,k2.Thus,L¯k1,k2g1,g2=L¯1□L¯2k1,k2g1,g2, for each k1,k2g1,g2∈ψ¯D1□ψ¯D22. Similarly, L¯=L¯1□L¯2. Hence, L=L1□L2 which clearly proves ℛ1□ℛ2L2=ℛ1L2□ℛ2L2.(2)
For any rough hypergraphℛ, ℛL2−1=ℛ,L is the rough hypergraph with a labeling function. The proof of part 2 is clear from part 1 and Remark 1. Using Remark 1, ℛ1□ℛ2L2−1=ℛ1□ℛ2,L. By proof of part 1, ℛ1□ℛ2,L=ℛ1□ℛ2,L1□L2=ℛ1,L1□ℛ2,L2=ℛ1L2−1□ℛ2L2−1. □Theorem 6.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2; then, for any k1,g1∈Q1 and k2,g2∈Q2,(12)dℛ1□ℛ2k1,k2,g1,g2=dℛ1k1,g1+dℛ2k2,g2.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 ℛ1 and ℛ2,(13)dℛ1□ℛ2k1,k2,g1,g2=dℛ1□ℛ22k1,k2,g1,g2=dℛ12□ℛ22k1,k2,g1,g2=dℛ12k1,g1+dℛ22k2,g2⇒dℛ1□ℛ2k1,k2,g1,g2=dℛ1k1,g1+dℛ2k2,g2.
Letℛ be a rough hypergraph. If there exist prime rough hypergraphs ℛ1,ℛ2,…,ℛq such that ℛ≅ℛ1□ℛ2□…□ℛq, then it is called PFD (prime factor decomposition) of ℛ into q 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.Algorithm 1: Method to compute PFD of a rough hypergraph.
(1)
Given a connected rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D(2)
Compute the L2-sectionℛ2=φ¯A,ψ¯D2,φ¯A,ψ¯D2 of ℛ(3)
Using the Algorithm of Imrich and Peterin [52], decompose the labeled graphs φ¯A,ψ¯D2 and φ¯A,ψ¯D2 into prime factors w.r.t the Cartesian product. That is, the edges of φ¯A,ψ¯D2 and φ¯A,ψ¯D2 are colored w.r.t the copies of corresponding prime factors.(4)
Merge the factors if necessary(5)
Computeℛ2−1 using the labeled and the PFD of ℛ is obtained such that the colored copies are the prime factorsDefinition 18.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2, respectively. A homomorphism from ℛ1 into ℛ2 is a mapping f:Q1⟶Q2 if there exist homomorphisms f¯:Q1⟶Q2 and f¯:Q1⟶Q2; that is,(1)
Iff¯D is a hyperedge in ℛ2¯, then D is a hyperedge in ℛ1¯(2)
Iff¯D is a hyperedge in ℛ2¯, then D is a hyperedge in ℛ1¯
A homomorphism which is a one-to-one correspondence betweenQ1 and Q2 is called an isomorphism. In this case, we say that the rough hypergraphs ℛ1 and ℛ2 are isomorphic to each other and write as ℛ1≅ℛ2.Definition 19.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D and ℛ′=φ¯A′,ψ¯D′,φ¯A′,ψ¯D′ be two rough hypergraphs on Q and Q′, respectively. A surjective homomorphism f:Q′⟶Q is called a covering projection if(1)
f¯−1k=f¯−1E¯=p, for all k∈φ¯A, E∈ψ¯D(2)
f¯−1k=f¯−1E¯=p, for all k∈φ¯A, E∈ψ¯D(3)
E¯′∩F¯′=∅, for all distinct E¯′,F¯′∈f−1E¯, E¯∈ψ¯D(4)
E¯′∩F¯′=∅, for all distinct E¯′,F¯′∈f−1E¯, E¯∈ψ¯D
The rough hypergraphℛ′ is called a p-fold covering of ℛ, and ℛ is called quotient rough hypergraph of ℛ′. If p=2, ℛ′ is called a double cover of ℛ.Definition 20.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The sets S¯⊆φ¯A and S¯⊆φ¯A are called independent if they contain no hyperedge of ℛ¯ and ℛ¯, respectively. The cardinalities of the largest independent sets are denoted by βℛ¯ and βℛ¯ and are called the independence number of ℛ¯ and ℛ¯, respectively. The value βℛ=1/2βℛ¯+βℛ¯ is called independence number of ℛ.Definition 21.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The subsets T¯⊆φ¯A and T¯⊆φ¯A are called covers of ℛ¯ and ℛ¯, respectively, if T¯∩E¯≠∅ and T¯∩E¯≠∅, for each E¯∈ψ¯D and E¯∈ψ¯D. The cardinalities of minimal covers are denoted by τℛ¯ and τℛ¯ and are called covering numbers of ℛ¯ and ℛ¯, respectively. The average value τℛ=1/2τℛ¯+τℛ¯ is called covering number of ℛ.Definition 22.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The fractional covers of ℛ¯ and ℛ¯ are, respectively, the mappings f¯:φ¯A⟶ℝ+∪0 and f¯:φ¯A⟶ℝ+∪0 such that(14)∑k∈E¯f¯k≥1,for eachE¯∈ψ¯D,∑k∈E¯f¯k≥1,for eachE¯∈ψ¯D.
The valueτ∗ℛ=1/2minf¯∑k∈φ¯Af¯k+minf¯∑k∈φ¯Af¯k is called fractional covering number (FC number) of ℛ.Definition 23.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The subsets M¯⊆ψ¯D and M¯⊆ψ¯D are called matching if every pair of hyperedges from M¯ and M¯ are mutually disjoint. The cardinalities of maximal matchings are denoted by νℛ¯ and νℛ¯ and are called matching numbers of ℛ¯ and ℛ¯, respectively. The matching number of ℛ is computed as νℛ=1/2νℛ¯+νℛ¯Definition 24.
Letℛ=φ¯A,ψ¯D,φ¯A,ψ¯D be a rough hypergraph on Q. The minimum number of mutually disjoint hyperedges whose union is the sets of vertices φ¯A and φ¯A is called the partition number of ℛ¯ and ℛ¯, respectively, denoted by ρℛ¯ and ρℛ¯. The value ρℛ=1/2ρℛ¯+ρℛ¯ is called the partition number of ℛ. If such partitions do not exist, then ρℛ=∞.
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. Letℛ1⊛ℛ2 denote any product of two rough hypergraphs ℛ1 and ℛ2. For any rough hypergraph ℛ, if there exists another rough hypergraph U such that ℛ⊛U=ℛ, then U is called the unit element. Note that U must be a hypergraph with a single vertex and no loops.Definition 25.
A rough hypergraphℛ=ℛ¯,ℛ¯ is called conformal if ℛ¯ and ℛ¯ are both conformal hypergraphs. That is, corresponding to each clique of 2-section ℛ¯2 (and ℛ¯2), there is a hyperedge in ℛ¯ (and ℛ¯).Definition 26.
Letℛ=ℛ¯,ℛ¯ be a rough hypergraph. The collection of all hyperedges in ℛ¯ (and ℛ¯) containing a common vertex k is called a star of ℛ¯ (and ℛ¯), denoted by ℛ¯k (and ℛ¯k). The pair ℛk=ℛ¯k,ℛ¯k is called a rough star of ℛ. The subsets E¯⊆ψ¯D and E¯⊆ψ¯D are called intersecting families of ℛ¯ and ℛ¯ if every pair of hyperedges of E¯ and E¯ have nonempty intersection. The pair E¯,E¯ is called a rough intersecting family of ℛ. A rough hypergraph ℛ is said to satisfy Helly property if each rough intersecting family in ℛ is a rough star.Definition 27.
Letℛ1′ and ℛ2′ be p1-fold and p2-fold coverings of rough hypergraphs ℛ1 and ℛ2 via covering projections f1=f¯1,f¯1 and f2=f¯2,f¯2, respectively; then, the Cartesian product f¯,f¯=f=f1□f2=f¯1□f¯2,f¯1□f¯2 is defined as(1)
f¯k1,k2=f¯1k1,f¯2k2, for all k1,k2∈φ¯A1×φ¯A2(2)
f¯k1,k2=f¯1k1,f¯2k2, for all k1,k2∈φ¯A1×φ¯A2(3)
f¯k1×E¯2=f¯1k1×f¯2E¯2, for all k1∈φ¯A1, E¯2∈ϕ¯D2(4)
f¯k1×E¯2=f¯1k1×f¯2E¯2, for all k1∈φ¯A1, E¯2∈ϕ¯D2(5)
f¯E¯1×k2=f¯1E¯1×f¯2k2, for all E¯1∈ϕ¯D1, k1∈φ¯A1(6)
f¯E¯1×k2=f¯1E¯1×f¯2k2, for all E¯1∈ϕ¯D1, k1∈φ¯A1Theorem 7.
Letℛ′1 and ℛ′2 be p1-fold and p2-fold coverings of rough hypergraphs ℛ1 and ℛ2 via covering projections f1 and f2, respectively; then, ℛ′1□ℛ′2 is a p1p2-fold covering of ℛ1□ℛ2 via covering projection f1□f2.Proof.
The mappingf=f1□f2 is given in Definition 27. We first need to show that f=f¯,f¯ is a surjective homomorphism. Let k1′×E¯′2 be a hyperedge in ℛ′1□ℛ′2; then, f¯k1′×E¯′2=f¯1k1′×f¯2E¯′2. Since f¯1 and f¯2 are homomorphisms, therefore f¯1k1′∈φ¯A2′ and f¯2E¯′2 is a hyperedge in ℛ¯2. Thus, f¯1k1′×f¯2E¯′2 is a hyperedge in ℛ¯1□ℛ¯2. Similarly, f¯E¯′1×k2′ is a hyperedge ℛ¯1□ℛ¯2 and f¯k1′×E¯′2; f¯E¯′1×k2′ are hyperedges in ℛ¯1□ℛ¯2 showing that f is a homomorphism. The surjectivity of f¯1□f¯2 and f¯1□f¯2 is obvious from the surjectivity of f¯1, f¯2, f¯1, and f¯2.
Letk1,k2∈φ¯A1×φ¯A2 be a vertex in ℛ¯1□ℛ¯2; then,(15)f¯1−1k1,k2=k1′,k2′∣f¯1k1′=k1,f¯2k2′=k2⇒f¯1−1k1,k2=f¯1−1k1×f¯2−1k2⇒f¯1−1k1,k2=f¯1−1k1×f¯2−1k2=p1p2.
Similarly,(16)f¯−1k1,k2=f¯1−1k1×f¯2−1k2=p1p2.
Consider a hyperedgek1×E¯2 in ℛ¯1□ℛ¯2; then,(17)f¯−1k1×E¯2=k1′,k2′|f¯1k1′=k1,f¯2k2′∈E¯2=f¯1−1k1×f¯2−1k2|f¯2k2′∈E¯2=f¯1−1k1×f¯2−1E¯2⇒f¯−1k1×E¯2=f¯1−1k1×f¯2−1E¯2=p1p2.
Similarly,(18)f¯1−1E¯1×k2=f¯1−1E¯1×f¯2−1k2=p1p2f¯−1k1×E¯2=f¯−1E¯1×k2=p1p2.
Letk1×E¯2 be a hyperedge in ℛ¯1□ℛ¯2 such that k1′×E¯′2,k1′×F¯′2∈f¯1−1k1×E¯2 and k1′×E¯′2∩k1′×F¯′2≠∅. It follows that E¯′2∩F¯′2≠∅, a contradiction. Similarly, we can prove the other cases. Hence, f=f¯,f¯ is covering projection and ℛ′1□ℛ′2 is a p1p2-fold covering of ℛ1□ℛ2. □Theorem 8.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1□ℛ2 is conformal if and only if ℛ1 and ℛ2 are conformal.Proof.
Letℛ1 and ℛ2 be two conformal rough hypergraphs. Let ℒ=ℒ¯,ℒ¯ be a rough clique in ℛ1□ℛ22; then, by Lemma 7, there are two possibilities.Case 1.
There exists a rough cliqueℒ1=ℒ¯1,ℒ¯1 in ℛ12 such that ℒ¯=ℒ¯1×x¯2, for some x¯2∈ψ¯D2, and ℒ¯=ℒ¯1×x¯2, for some x¯2∈ψ¯D2. Since ℒ is a rough clique, therefore, ℒ1 is also a rough clique. As ℛ1 is a conformal rough hypergraph, therefore there exist hyperedges E¯1 and E¯1 corresponding to ℒ¯1 and ℒ¯1 in ℛ¯1 and ℛ¯1. Thus, E¯1×x¯2,E¯1×x¯2 is a rough hyperedge corresponding to ℒ1 showing that ℛ1□ℛ2 is a conformal rough hypergraph.Case 2.
There exists a rough cliqueℒ2=ℒ¯2,ℒ¯2 in ℛ22 such that ℒ¯=x¯1×ℒ¯2, for some x¯1∈ψ¯D1, and ℒ¯=x¯1×ℒ¯2, for some x¯1∈ψ¯D1. This case can be proved along the same lines as Case 1. Hence, ℛ1□ℛ2 is a conformal rough hypergraph.
Conversely, letℛ1□ℛ2 be a conformal rough hypergraph. Let ℒ1 and ℒ2 be rough cliques in ℛ1 and ℛ2, respectively. By Lemma 7, ℒ=ℒ1□ℒ2 is a rough clique in ℛ1□ℛ22. Since ℛ1□ℛ2 is a conformal rough hypergraph, therefore, there exists a rough hyperedge E=E¯,E¯ in ℛ1□ℛ2 corresponding to ℒ. By Lemma 7, there exist rough hyperedges E1 and E2 in ℛ1 and ℛ2 such that E=E1□E2. Clearly, E1 and E2 correspond to in ℒ1 and ℒ2, respectively, proving that ℛ1□ℛ2 are conformal rough hypergraphs. □Theorem 9.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1□ℛ2 has the Helly property iff ℛ1 and ℛ2 have the Helly property.Definition 28.
The dual of a rough hypergraphℛ=φ¯A,ψ¯D,φ¯A,ψ¯D is a rough hypergraph ℛ∗=φ¯A∗,ψ¯D∗,φ¯A∗,ψ¯D∗, where(1)
The hyperedge setψ¯D of ℛ is the vertex set of ℛ∗, that is, φ¯A∗=ψ¯D=E¯1,E¯2,…,E¯r(2)
The edge setψ¯D of ℛ is the vertex set of ℛ∗, that is, φ¯A∗=ψ¯D=E¯1,E¯2,…,E¯r(3)
If|φ¯A|=m, then ψ¯D∗=K¯1,K¯2,…,K¯m is the hyperedge set of ℛ∗ such that K¯i=E¯j|ki∈E¯j, that is, K¯i is the collection of those hyperedges of ℛ¯ which share the common vertex ki(4)
Ifφ¯A=n, then ψ¯D∗=K¯1,K¯2,…,K¯n is the hyperedge set of ℛ∗ such that K¯i=E¯j|ki∈E¯j, that is, K¯i is the collection of those hyperedges of ℛ¯ which share the common vertices kiRemark 3.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1□ℛ2∗ may not be equal to ℛ1∗□ℛ2∗. Since, for any two hypergraphs H1 and H2, H1□H2∗ is not equal to h1∗□H2∗ 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 thatH1□H2 has nine edges and so H1□H2∗ has nine vertices. But, H1∗□H2∗ has three vertices. The vertex sets of H1□H2∗ and H1∗□H2∗ are not equal and it proves our claim.Figure 7
HypergraphH1.Figure 8
HypergraphH2.


## 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.
Letℛ1∗ℛ2∗⋯∗ℛr be any product of rough hypergraphs and Vℛi denotes the vertex set of ℛi, for any 1≤i≤r. The mapping pi=p¯i,p¯i:Vℛ1∗ℛ2∗⋯∗ℛr⟶Vℛi is called the projection of ℛ1∗ℛ2∗⋯∗ℛr onto ith factor ℛi, where p¯i:ℛ¯1∗ℛ¯2∗⋯∗ℛ¯r⟶ℛ¯i and p¯i:ℛ¯1∗ℛ¯2∗⋯∗ℛ¯r⟶ℛ¯i are the projection mappings defined as(19)p¯ik1,k2,…,kn=p¯ik1,k2,…,kn=ki,for allki∈Qi,1≤i≤r.Definition 30.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The square product of ℛ1 and ℛ2 is a rough hypergraph ℛ1×ℛ2=ℛ¯1×ℛ¯2,ℛ¯1×ℛ¯2 which is defined as(1)
ℛ¯1×ℛ¯2=φ¯A1×φ¯A2,ψ¯D1×ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×ψ¯D2=E¯1×E¯2|E¯1∈ψ¯D1,E¯2∈ψ¯D2(2)
ℛ¯1□ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×ψ¯D2=E¯1×E¯2|E¯1∈ψ¯D1,E¯2∈ψ¯D2
In short,ℛ¯1×ℛ¯2 and ℛ¯1×ℛ¯2 are the square products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1×ℛ2×ℛ3≅ℛ1×ℛ2×ℛ3(2)
ℛ1×ℛ2≅ℛ2×ℛ1(3)
ℛ1×ℛ2∪ℛ3≅ℛ1×ℛ2∪ℛ1×ℛ2(4)
ℛ×U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1×ℛ2⟶ℛ1 and p2:Vℛ1×ℛ2⟶Vℛ2 are homomorphismsDefinition 31.
Letℛ be a rough hypergraph; then, ℛ is called an r-uniform rough hypergraph if ℛ¯ and ℛ¯ are both r-uniform hypergraphs; that is, for each E¯∈ψ¯D and E¯∈ψ¯D, E¯=E¯=r.Lemma 7.
Letℛ1 and ℛ2 be two r-uniform rough hypergraphs; then, ℛ1×ℛ22≅ℛ12×ℛ22.Proof.
Since the vertex set ofℛi and ℛi2 is the same for i=1,2, therefore, the vertex set of ℛ1×ℛ22 and ℛ12×ℛ22 is the same. It only needs to show that the set of hyperedges of ℛ1×ℛ22 and ℛ12×ℛ22 is the same. By Definition 30, ℛ1×ℛ22=ℛ¯1×ℛ¯22,ℛ¯1×ℛ¯22, where ℛ¯1×ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×ψ¯D22 and ℛ¯1×ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×ψ¯D22. As ℛ1 and ℛ2 are r-uniform rough hypergraphs, so we have(20)ψ¯D1×ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1×ψ¯D2=x1,x2x1,y2|x1,y1⊆E¯1∈ψ¯D1,x2,y2⊆E¯2∈ψ¯D2,E¯1=E¯2=x1,x2x1,y2|x1y1∈ψ¯D12,x2y2∈ψ¯D22⇒ψ¯D1×ψ¯D22=ψ¯D12×ψ¯D22.
As the vertex setℛ¯1×ℛ¯22 and ℛ¯12×ℛ¯22 is also the same, so ℛ¯1×ℛ¯22≅ℛ¯12×ℛ¯22. Similarly, ℛ¯1×ℛ¯22≅ℛ¯12×ℛ¯22. Hence, ℛ1×ℛ22≅ℛ12×ℛ22.Theorem 10.
Letℛ1 and ℛ2 be two rough hypergraphs; then,(1)
rℛ1×ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(2)
sℛ1×ℛ2=maxsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Proof.
Since, by Definition30, ℛ1×ℛ2=ℛ¯1×ℛ¯2,ℛ¯1×ℛ¯2, we first need to compute rℛ¯1×ℛ¯2 and rℛ¯1×ℛ¯2. By Definition 6,(21)rℛ¯1×ℛ¯2=maxE¯∈ψ¯D1×ψ¯D2E¯=maxE¯|p1E¯=E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=maxmaxE¯1∈ψ¯D1E¯1,maxE¯2∈ψ¯D2E¯2⇒rℛ¯1×ℛ¯2=maxrℛ¯1,rℛ¯2.
Similarly,rℛ¯1×ℛ¯2=maxrℛ¯1,rℛ¯2. Hence,(22)rℛ1×ℛ2=12maxrℛ¯1,rℛ¯2+maxrℛ¯1,rℛ¯2=maxrℛ¯1+rℛ¯12,rℛ¯2+rℛ¯22,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12⇒rℛ¯1×ℛ¯2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12.
Using Definition6, the antirank of ℛ1×ℛ2 is given as(23)sℛ¯1×ℛ¯2=minE¯∈ψ¯D1×ψ¯D2E¯=minE¯|p1E¯=E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=maxminE¯1∈ψ¯D1E¯1,minE¯2∈ψ¯D2E¯2⇒sℛ¯1×ℛ¯2=maxsℛ¯1,sℛ¯2.
Similarly,sℛ¯1×ℛ¯2=maxsℛ¯1,sℛ¯2. Hence,(24)sℛ1×ℛ2=12maxsℛ¯1,sℛ¯2+maxsℛ¯1,sℛ¯2=maxsℛ¯1+sℛ¯12,sℛ¯2+sℛ¯22,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12⇒sℛ1×ℛ2=maxsℛ1,sℛ2,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12.


## 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The direct product ×⌣ of ℛ1 and ℛ2 is a rough hypergraph ℛ1×⌣ℛ2=ℛ¯1×⌣ℛ¯2,ℛ¯1×⌣ℛ¯2 which is defined as(1)
ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×⌣ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯∈ψ¯D2(2)
ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1×⌣ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯∈ψ¯D2
In short,ℛ¯1×⌣ℛ¯2 and ℛ¯1×⌣ℛ¯2 are the MRP direct products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1×⌣ℛ2×⌣ℛ3≅ℛ1×⌣ℛ2×⌣ℛ3(2)
ℛ1×⌣ℛ2≅ℛ2×⌣ℛ1(3)
ℛ1×⌣ℛ2∪ℛ3≅ℛ1×⌣ℛ2∪ℛ1×⌣ℛ2(4)
ℛ×⌣U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1×⌣ℛ2⟶ℛ1 and p2:Vℛ1×⌣ℛ2⟶Vℛ2 may not be weak homomorphismsLemma 8.
Letℛ1 and ℛ2 be two rough hypergraphs; then, ℛ1×⌣ℛ22≅ℛ12□ℛ22∪ℛ12×ℛ22.Proof.
Since the vertex set ofℛi and ℛi2 is the same for i=1,2, therefore, the vertex set of ℛ1×⌣ℛ22 and the union of ℛ12□ℛ22 and ℛ12×ℛ22 is the same. It only needs to show that the set of hyperedges of ℛ1×⌣ℛ22 and ℛ12□ℛ22∪ℛ12×ℛ22 is the same. By Definition 32, ℛ1×⌣ℛ22=ℛ¯1×⌣ℛ¯22,ℛ¯1×⌣ℛ¯22, where ℛ¯1×⌣ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D22 and ℛ¯1×⌣ℛ¯22=φ¯A1×φ¯A2,ψ¯D1×⌣ψ¯D22. So, we have(25)ψ¯D1×⌣ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1×ψ¯D2ψ¯D1×⌣ψ¯D22=x1,x2y1,y2|x1,x2,y1,y2⊆E¯∈ψ¯D1×ψ¯D2=x1,x2y1,y2|x1=y1,x2,y2⊆p2E¯∈ψ¯D2∪x1,x2x1,y2|x1,y1⊆p1E¯∈ψ¯D1,x2=y2∪x1,x2x1,y2∣x1,y1⊆p1E¯∈ψ¯D1,x2,y2⊆p2E¯∈ψ¯D2,E¯1=E¯2=x1,x2x1,y2|x1,y1∈φ¯A1,x2y2∈ψ¯D22x1,x2x1,y2|x1y1∈ψ¯D12,x2,y2∈φ¯A2x1,x2x1,y2|x1y1∈ψ¯D12,x2y2∈ψ¯D22⇒ψ¯D1×⌣ψ¯D22=ψ¯D12□ψ¯D22∪ψ¯D12×ψ¯D22.
Thus,ℛ¯1×⌣ℛ¯22≅ℛ¯12□ℛ¯22∪ℛ¯12×ℛ¯22. Similarly, ℛ¯1×⌣ℛ¯22≅ℛ¯12□ℛ¯22∪ℛ¯12×ℛ¯22. Hence, ℛ1×ℛ22≅ℛ12□ℛ22∪ℛ12×ℛ22.Theorem 11.
Letℛ1 and ℛ2 be two rough hypergraphs; then,(1)
rℛ1×⌣ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(2)
sℛ1×⌣ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Proof.
Since, by Definition32, ℛ1×⌣ℛ2=ℛ¯1×⌣ℛ¯2,ℛ¯1×⌣ℛ¯2, we first need to compute rℛ¯1×⌣ℛ¯2 and rℛ¯1×⌣ℛ¯2. By Definition 6,(26)rℛ¯1×⌣ℛ¯2=maxE¯∈ψ¯D1×⌣ψ¯D2E¯=maxE¯|p1E¯=E¯1∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=maxmaxE¯1∈ψ¯D1E¯1,maxE¯2∈ψ¯D2E¯2⇒rℛ¯1×⌣ℛ¯2=maxrℛ¯1,rℛ¯2.
Similarly,rℛ¯1×⌣ℛ¯2=maxrℛ¯1,rℛ¯2. Hence,(27)rℛ1×⌣ℛ2=12maxrℛ¯1,rℛ¯2+maxrℛ¯1,rℛ¯2=maxrℛ¯1+rℛ¯12,rℛ¯2+rℛ¯22,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12⇒rℛ1×⌣ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯22,rℛ¯2+rℛ¯12.
Using Definition6, the antirank of ℛ1×⌣ℛ2 is given as(28)sℛ¯1×⌣ℛ¯2=minE¯∈ψ¯D1×⌣ψ¯D2E¯=minE¯|p1E¯=E¯1∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪E¯|p1E¯⊆E¯1∈ψ¯D1,p2E¯=E¯2∈ψ¯D2=minminE¯1∈ψ¯D1E¯1,minE¯2∈ψ¯D2E¯2⇒sℛ¯1×⌣ℛ¯2=minsℛ¯1,sℛ¯2.
Similarly,sℛ¯1×⌣ℛ¯2=maxsℛ¯1,sℛ¯2. Hence,(29)sℛ1×⌣ℛ2=12minsℛ¯1,sℛ¯2+minsℛ¯1,sℛ¯2=minsℛ¯1+sℛ¯12,sℛ¯2+sℛ¯22,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12⇒sℛ1×⌣ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯22,sℛ¯2+sℛ¯12.


## 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The union f ℛ1 and ℛ2 is a rough hypergraph ℛ1∪ℛ2=ℛ¯1∪ℛ¯2,ℛ¯1∪ℛ¯2, where ℛ¯1∪ℛ¯2=φ¯A1∪φ¯A2,ψ¯D1∪ψ¯D2 and ℛ¯1∪ℛ¯2=φ¯A1∪φ¯A2,ψ¯D1∪ψ¯D2.
The intersectionfℛ1 and ℛ2 is a rough hypergraph ℛ1∩ℛ2=ℛ¯1∩ℛ¯2,ℛ¯1∩ℛ¯2, where ℛ¯1∩ℛ¯2=φ¯A1∩φ¯A2,ψ¯D1∩ψ¯D2 and ℛ¯1∩ℛ¯2=φ¯A1∩φ¯A2,ψ¯D1∩ψ¯D2.
In short,ℛ¯1∪ℛ¯2 and ℛ¯1∩ℛ¯2 are the union and intersection of lower approximate hypergraphs ℛ¯1 and ℛ¯2. 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 ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1∪ℛ2∪ℛ3≅ℛ1∪ℛ2∪ℛ3(2)
ℛ1∩ℛ2∩ℛ3≅ℛ1∩ℛ2∩ℛ3(3)
ℛ1∪ℛ2≅ℛ2∪ℛ1(4)
ℛ1∩ℛ2≅ℛ2∩ℛ1(5)
ℛ1∪ℛ2∩ℛ3≅ℛ1∪ℛ2∩ℛ1∪ℛ2(6)
The projectionsp1:Vℛ1∪ℛ2⟶ℛ1 and p2:Vℛ1∪ℛ2⟶Vℛ2 may not be weak homomorphisms(7)
The projectionsp1:Vℛ1∩ℛ2⟶ℛ1 and p2:Vℛ1∩ℛ2⟶Vℛ2 may not be weak homomorphismsRemark 4.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs, then(1)
ℛ1∪ℛ22≅ℛ12∪ℛ22(2)
rℛ1∪ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(3)
sℛ1∪ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2(4)
ℛ1∪ℛ2∗≅ℛ1∗∪ℛ2∗


## 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The strong product of ℛ1 and ℛ2 is a rough hypergraph ℛ1⊠ℛ2=ℛ¯1⊠ℛ¯2,ℛ¯1⊠ℛ¯2, where ℛ¯1⊠ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×ψ¯D2 and ℛ¯1∪ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×ψ¯D2. In other words, the strong product of ℛ1 and ℛ2 is the union of Cartesian product and square product of rough hypergraphs ℛ1 and ℛ2.
In short,ℛ¯1⊠ℛ¯2 and ℛ¯1⊠ℛ¯2 are the strong products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1⊠ℛ2⊠ℛ3≅ℛ1⊠ℛ2⊠ℛ3(2)
ℛ1⊠ℛ2≅ℛ2⊠ℛ1(3)
ℛ1⊠ℛ2∪ℛ3≅ℛ1⊠ℛ2∪ℛ1⊠ℛ2(4)
ℛ⊠U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1⊠ℛ2⟶ℛ1 and p2:Vℛ1⊠ℛ2⟶Vℛ2 may not be weak homomorphisms(6)
ℛ1⊠ℛ22≅ℛ12⊠ℛ22(7)
rℛ1⊠ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(8)
sℛ1⊠ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Theorem 12.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2; then, for any k1,g1∈Q1 and k2,g2∈Q2,(30)dℛ1⊠ℛ2k1,k2,g1,g2=dℛ1k1,g1,dℛ2k2,g2,dℛ¯1k1,g1+dℛ¯2k2,g22,dℛ¯1k1,g1+dℛ¯2k2,g22.Proof.
The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2, and the result ℛ1⊠ℛ22≅ℛ12⊠ℛ22. Thus, for any two rough hypergraphs ℛ1 and ℛ2,(31)dℛ¯1⊠ℛ¯2k1,k2,g1,g2=dℛ¯1⊠ℛ¯22k1,k2,g1,g2=dℛ¯12⊠ℛ¯22k1,k2,g1,g2=maxdℛ¯12k1,g1,dℛ¯22k2,g2⇒dℛ¯1⊠ℛ¯2k1,k2,g1,g2=maxdℛ¯1k1,g1,dℛ¯2k2,g2.
Similarly,dℛ¯1⊠ℛ¯2k1,k2,g1,g2=maxdℛ¯1k1,g1,dℛ¯2k2,g2, 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The strong product of ℛ1 and ℛ2 is a rough hypergraph ℛ1⊠⌣ℛ2=ℛ¯1⊠⌣ℛ¯2,ℛ¯1⊠⌣ℛ¯2, where ℛ¯1⊠⌣ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×⌣ψ¯D2 and ℛ¯1⊠⌣ℛ¯2=ℛ¯1□ℛ¯2∪ℛ¯1×⌣ℛ¯2=φ¯A1×φ¯A2,ψ¯D1□ψ¯D2∪ψ¯D1×⌣ψ¯D2. In other words, the strong product of ℛ1 and ℛ2 is the union of Cartesian product and direct product of rough hypergraphs ℛ1 and ℛ2.
In short,ℛ¯1⊠⌣ℛ¯2 and ℛ¯1⊠⌣ℛ¯2 are the strong products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1⊠⌣ℛ2⊠⌣ℛ3≅ℛ1⊠⌣ℛ2⊠⌣ℛ3(2)
ℛ1⊠⌣ℛ2≅ℛ2⊠⌣ℛ1(3)
ℛ1⊠⌣ℛ2∪ℛ3≅ℛ1⊠⌣ℛ2∪ℛ1⊠⌣ℛ2(4)
ℛ⊠⌣U≅ℛ, where U is a single vertex hypergraph without loops(5)
The projectionsp1:Vℛ1⊠⌣ℛ2⟶ℛ1 and p2:Vℛ1⊠⌣ℛ2⟶Vℛ2 may not be weak homomorphisms(6)
ℛ1⊠⌣ℛ22≅ℛ12⊠ℛ22(7)
rℛ1⊠⌣ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(8)
sℛ1⊠⌣ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Theorem 13.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2, then for any k1,g1∈Q1 and k2,g2∈Q2,(32)dℛ1⊠⌣ℛ2k1,k2,g1,g2=dℛ1k1,g1,dℛ2k2,g2,dℛ¯1k1,g1+dℛ¯2k2,g22,dℛ¯1k1,g1+dℛ¯2k2,g22.Proof.
The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2 and the result ℛ1⊠⌣ℛ22≅ℛ12⊠ℛ22. Thus, for any two rough hypergraphs ℛ1 and ℛ2. □


## 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.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The lexicographic product ° of ℛ1 and ℛ2 is a rough hypergraph ℛ1°ℛ2=ℛ¯1°ℛ¯2,ℛ¯1°ℛ¯2 which is defined as(1)
ℛ¯1°ℛ¯2=φ¯A1×φ¯A2,ψ¯D1°ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1°ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪k1×E¯2|k1∈ψ¯D1,E¯2∈ψ¯D2(2)
ℛ¯1°ℛ¯2=φ¯A1×φ¯A2,ψ¯D1°ψ¯D2(a)
φ¯A1×φ¯A2=k1,k2|k1∈φ¯A1,k2∈φ¯A2(b)
ψ¯D1°ψ¯D2=E¯|p1E¯∈ψ¯D1,p2E¯⊆E¯2∈ψ¯D2∪k1×E¯2|k1∈ψ¯D1,E¯2∈ψ¯D2
In short,ℛ¯1°ℛ¯2 and ℛ¯1°ℛ¯2 are the lexicographic products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U (left identity) as a trivial hypergraph with a single vertex such that U¯=U¯. That is, for any rough hypergraphs ℛ,ℛ1, ℛ2, and ℛ3, the following properties hold:(1)
ℛ1°ℛ2°ℛ3≅ℛ1°ℛ2°ℛ3(2)
ℛ1°ℛ2≅ℛ2°ℛ1(3)
ℛ1°ℛ2∪ℛ3≅ℛ1°ℛ2∪ℛ1°ℛ2(4)
U°ℛ≅ℛ, where U is a single vertex hypergraph without loops(5)
ℛ°U≅ℛ(6)
The projectionsp1:Vℛ1°ℛ2⟶ℛ1 and p2:Vℛ1°ℛ2⟶Vℛ2 may not be weak homomorphisms(7)
ℛ1°ℛ22≅ℛ12°ℛ22(8)
rℛ1°ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(9)
sℛ1°ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2Theorem 14.
Letℛ1 and ℛ2 be two rough hypergraphs on Q1 and Q2; then, for any k1,g1∈Q1 and k2,g2∈Q2,(33)dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2,ifk1=g1,degk1≠0,dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2,ifk1=g1,degk1≠0.Proof.
The proof of this theorem is a direct consequence of Proposition 5.4 of [5], Lemma 2, and the result ℛ1⊠ℛ22≅ℛ12⊠ℛ22. Thus, for any two rough hypergraphs ℛ1 and ℛ2,(34)dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1°ℛ¯22k1,k2,g1,g2=dℛ¯12°ℛ¯22k1,k2,g1,g2=dℛ¯12k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2ifk1=g1,degk1≠0,⇒dℛ¯1°ℛ¯2k1,k2,g1,g2=dℛ¯1k1,g1,ifk1≠g1,dℛ¯2k2,g2,ifk1=g1,degk1=0,mindℛ¯2k2,g2,2,ifk1=g1,degk1≠0.
Similarly for upper approximate hypergraphs, the result follows.Definition 37.
Letℛ1=ℛ¯1,ℛ¯1 and ℛ2=ℛ¯2,ℛ¯2 be two rough hypergraphs. The costrong product ⋆ of ℛ1 and ℛ2 is a rough hypergraph ℛ1⋆ℛ2=ℛ¯1⋆ℛ¯2,ℛ¯1⋆ℛ¯2 which is defined as ℛ1⋆ℛ2=ℛ1°ℛ2∪ℛ2°ℛ1.
In short,ℛ¯1°ℛ¯2 and ℛ¯1°ℛ¯2 are the costrong products of lower approximate hypergraphs ℛ¯1,ℛ¯2 and upper approximate hypergraphs ℛ¯1,ℛ¯2, 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 U (left identity) as a trivial hypergraph with a single vertex such that U¯=U¯.Remark 5.
Letℛ1 and ℛ2 be two rough hypergraphs, then(1)
ℛ1⋆ℛ22=ℛ12⋆ℛ22(2)
rℛ1⋆ℛ2=maxrℛ1,rℛ2,rℛ¯1+rℛ¯2/2,rℛ¯2+rℛ¯1/2(3)
sℛ1⋆ℛ2=minsℛ1,sℛ2,sℛ¯1+sℛ¯2/2,sℛ¯2+sℛ¯1/2


## 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.



---

*Source: 1540004-2022-03-07.xml*