Decision Making Based on Valued Fuzzy Superhypergraphs

1  Introduction

The theory of hypergraph as a model of hypernetwork has been introduced by Berge as a generalization of graph theory in 1960 [1]. A graph cannot connect more than two elements, which is a defect when discussing the connection between a group of elements is considered. A hypergraph is a generalization of a graph such that its edge can relate any number of vertices, in which a graph, and its edge connects exactly two vertices. Indeed, a hypergraph is a useful tool to analyze the structure of a system that connects a group of elements (more than two elements) together, and in this sense, it can play an important role in embedding, classifying, partitioning, covering, and clustering elements in different classes. Therefore, a hypergraph as an extension and covering of the graph has attracted the attention of many researchers, and this theory has spread rapidly, especially since it has many applications in real world (its uses in model gene interactions (ER-type hypergraph [2]), machine learning (spectral hypergraph [3]), computer networks (WIS hypergraph [4]), chemistry (molecular hypergraph [5]), visual classification (hypergraph-induced convolutional network [6]), and social media (soft hypergraph [7])). Because of the more important and updated hypergraphs, a hypergraph structure as a hypernetwork has various applications, among which it can refer to updated works such as soft hypergraph for modeling global interactions via social media networks [7], session-based recommendation with hypergraph convolutional networks and sequential information embeddings [8], hypergraph-based analysis and design of intelligent collaborative manufacturing space [9] and hypergraph-based centrality metrics for maritime container service networks: a worldwide application [10]. In classical set theory, the mathematical concepts introduce purely, and without any quality or criteria, it is not attractive to be used in the world. Zadeh introduced the concept of fuzzy set theory as a generalization of set theory to deal with uncertainties [11]. Furthermore, the Plithogenic set (as a generalization of crisp, fuzzy, Intuitionistic fuzzy, and neutrosophic sets) was introduced by Smarandache in 2017. The Plithogenic set is a set whose elements are characterized by attribute values. Recently, Edalatpanah et al. considered new and recent developments in methodologies, techniques, and applications of Neutrosophic and Plithogenic sets for various practical problems and demonstrated the challenging issues as a generalization of fuzzy sets [12]. Sometimes graphs are not able to correctly analyze many phenomena because the uncertainty of various characteristics of systems naturally exists, so fuzzy graphs can cover this defect and this is an important motivation to introduce fuzzy graph theory. After that, the fuzzy graph theory was proposed as a generalization of the graph by Rosenfeld [13], which has many applications in real world (see more details in [14]). Since a fuzzy graph as a complex graph theory gives very limited information about complex networks, so we propose that the main motivation of complex hypergraph structures is for covering fuzzy graph defects in applications. Lee-Kwang et al. [15], generalized and redefined the concept of fuzzy hypergraphs (as a useful tool for the analysis and fuzzy partition of a system) whose basic idea was given by Kaufmann [16]. Akram et al. have written a book on fuzzy hypergraph which presents the fundamental and technical concepts of fuzzy hypergraphs and explains their extensions and applications. It discusses applied generalized mathematical models of hypergraphs, including complex, intuitionistic, bipolar, m-polar fuzzy, Pythagorean, complex Pythagorean, and q-rung ortho-pair hypergraphs, as well as single-valued neutrosophic, complex neutrosophic, and bipolar neutrosophic hypergraphs. In addition, the book also sheds light on real-world applications of these hypergraphs, making it a valuable resource for students and researchers in the field of mathematics, as well as computer and social scientists [17]. There is also some research about fuzzy (hyper) graphs and their applications in complex hypernetworks, such as the implementation of single-valued neutrosophic soft hypergraphs on the human nervous system [18], decision-making methods based on fuzzy soft competition hypergraphs [19], hypergraph and network flow-based quality function deployment [20], global domination in fuzzy graphs using strong arcs [21], fuzzy hypergraph modeling, analysis and prediction of crimes [22], single-valued neutrosophic directed (hyper) graphs and applications in networks [23], achievable single-valued neutrosophic graphs in wireless sensor networks [24], fuzzy hypergraph network for recommending top-k profitable stocks [25], an algorithm to compute the strength of competing interactions in the bearing sea based on Pythagorean fuzzy hypergraphs [26] and centrality measures in fuzzy social networks [27]. Recently, Smarandache extended hypergraphs to a new concept as n-superhypergraph and Plithogenic n-superhypergraph which have several properties and are connected with the real-world [28]. Indeed, n-superhypergraphs are a generalization of hypergraphs, with the advantage that they can communicate between the hyperedges. Recently, Hamidi et al. introduced the concept of quasi superhypergraphs as a special concept of n-superhypergraphs and showed that in hypergraph theory, any hypergraph can relate a set of elements, while without any details, it makes some conflicts, defects, and shortcomings in hypergraph theory [29]. Thus, by introducing superhypergraph, they try to eliminate defects of graphs (sometimes graph structures give very limited information about complex networks) structures and hypergraph structures (although the hypergraph structures are for covering graph defects in the applications in hypergraphs, the relation between vertices can’t be described in full details). They introduced the incidence matrix of superhypergraph and computed the characteristic polynomial for the incidence matrix of superhypergraph, so they obtained the spectrum of superhypergraphs. Also, by computingthe number of superedges of any given superhypergraphs and based on superedges and partitions of an underlying set of superhypergraph, the number of all superhypergraphs on any nonempty set is obtained.

Regarding these points, we introduce the concept of valued fuzzy superhypergraphs as a generalization of fuzzy hypergraphs. Valued fuzzy superhypergraphs are dependent on the concept of fuzzy supervertices and fuzzy superedges or fuzzy links, which are defined in this study. The motivation of valued fuzzy superhypergraphs is based on a design of real problems as complex superhypernetworks. Indeed, we modify a real problem as a fuzzy superhypergraph, and with respect to the notation of the impact membership value of modeled fuzzy superhypergraph, we have made the best decision. This paper presents the fuzzy algebraic structures on valued fuzzy superhypergraphs such as strong valued fuzzy superhypergraphs, isomorphic valued fuzzy superhypergraphs, the complement of fuzzy link, and self-complemented valued fuzzy superhypergraphs. Also, the relation between fuzzy hypergraphs and strongly valued fuzzy superhypergraphs has been investigated.

Motivation and advantage: Modeling based on fuzzy (hyper) graphs is a clustering or grouping of elements based on certain properties, in which the properties of its elements are checked in each cluster, and this check has nothing to do with the properties of other clusters. We need more complete modeling in order to be able to analyze the effect of the elements in the entire modeling and with other cluster elements at the same time. Our motivation in introducing valued fuzzy superhypergraphs is the coverage of this problem in fuzzy (hyper) graphs and therefore we define it such a way that in each cluster and group the elements are related to the elements of other clusters and groups in order to analyze the effect of each element in the (super) hypernetwork on the whole (super) hypernetwork. One of the motivations for valued fuzzy superhypergraphs is that we can have the best modeling of complex (super) hypernetworks where the role of all the details that make up this complex (super) hypernetwork is important in detail and in general. In fact, in this modeling, for each complex (super) hypernetwork, we examine the relationship between all the components of the systems that make up the complex (super) hypernetwork in detail and analyze the impact of these components in the entire complex (super) hypernetwork. The highest advantage of modeling the world’s issues based on valued fuzzy superhypergraph regarding the modeling of the fuzzy (hyper) graph is in the complex (super) hypernetworks based on the fuzzy superhypergraph, and the relationship between the components is checked only in the components of all clusters based on the map of clusters. This advantage leads to obtaining, the optimal results in the complex (super) hypernetwork, based on our mathematical methods and computations. In this method, we consider the values of components in clusters, the values between clusters, and the weight of clusters and closely and carefully find the suitable positive real values for optimal computations in complex (super) hypernetworks. The positive real values play the main role in the extremum in calculus and so the optimization problem plays a significant role in these calculations.

2  Preliminaries

In this section, we recall some definitions and results, which we need as follows:

Definition 2.1. [1] Let X be a finite set. A hypergraph on X is a pair H=(X,{Ei}i=1m) such that for all 1≤i≤m,∅≠Ei⊆X and ⋃i=1mEi=X. The elements x1,x2,…,xn of X are called vertices, and the sets E1,E2,…,Em are called the hyperedges of the hypergraph H. In hypergraphs, hyperedges can contain an element (loop) two elements (edge) or more than three elements. A hypergraph H=(X,{Ei}i=1m) is called a complete hypergraph, if for any x, y ∈ X, there is 1 ≤ i ≤ m such that {x,y}⊆Ei. A hypergraph H=(X,{Ei}i=1n) is called a joint complete hypergraph, if |X| = n for all 1≤i≤n,|Ei|=i and Ei⊆Ei+1 element (loop). If for all 1 ≤ k ≤ m |Ek|=2, the hypergraph becomes an ordinary (undirected) graph. and n rows representing the vertices x1,x2,…,xn, where for all 1 ≤ i ≤ n and for all 1 ≤ j ≤ m, mij=1 if xi∈Ej and mij=0 if xi∉Ej.

Definition 2.2. [30] Let m∈N and V={v1,v2,…,vm} be a set of vertices, that contain single vertices (the classical ones), indeterminant vertices (unclear, vague, unknown) and null vertices (unknown, empty). Consider P(V) as the power set of V, P2(V)=P(P(V))…, and Pn+1(V)=P(Pn(V)) be the n-power set of the set V. Then the n-superhypergraph (n-SHG) is an ordered pair n-SHG=(Gn,En), where for any n∈N, Gn⊆Pn(V), is the set of vertices and En⊆Pn(V) is the set of edges. The set Gn contains some type of vertices, such as single vertices (the classical ones), indeterminate vertices (unclear, vague, partially unknown), null vertices (totally unknown, empty), supervertices (or subset vertex), i.e., two or more (single, indeterminate, or null) vertices that together from a group (organization). An n-supervertex is a collection of many vertices such that at least one is a (n − 1)-supervertex and all other supervertices in the collection have the order r ≤ n − 1. The set of edges En contains some edges such as single edges (the classical ones), indeterminant (unclear, vague, partially unknown), null-edge (empty, totally unknown), hyperedge (containing three or more single vertices), superedge (containing two vertices, at least one of them being a supervertex), n-superedge (containing two vertices, at least one being an n-supervertex and the other of order r-supervertex with r ≤ n), superhyperedge (containing three or more vertices, at least one being a supervertex), n-superhyper edge (containing three or more vertices, at least one being an n-supervertex and the other r-supervertices with r ≤ n), multi edges (two or more edges connecting the same two vertices), and loop (an edge that connects an element).

Definition 2.3. [29] Let X be a non-empty set. Then

(i)   H=(X,S={Si}i=1k,Φ={φi,j}i,j) is called a quasi superhypergraph, if Φ≠∅ and X=⋃i=1nSi, where k ≥ 2, and for all 1≤i≤k,Si∈P∗(X), is called a supervertex and for any i≠j, the map φi,j:Si→Sj (say Si links to Sj) is called a superedge,

(ii)   the quasi superhypergraph H=(X,{Si}i=1k,{φi,j}i,j) is called a superhypergraph, if for any Si∈P∗(X) there exists at least one Sj∈P∗(X) such that Si links to Sj (it is not necessary all supervertices be linked).

(iii)   The superhypergraph H=(X,{Si}i=1k,{φi,j}i,j) is called a trivial superhypergraph, if k = 1 (S1 can’t link to itself).

Let X be a nonempty set, 𝒮ℋ(X)={H|His a superhypergraph onX} and 𝒮ℋ(n1,n2,…,nk)={(X,{Si}i=1k,{φi,j})∈𝒮ℋ||Si|=ni}.

Theorem 2.1. [29] Let X be a nonempty set and |X| = n. If α={(n1,n2,…,nr)|∑i=1rni=n,ni,r∈N} and m=|{i|ni=nj}|, then |𝒮ℋ(X)|=∑α(1m!∏i=1r(n−∑j=1i−1njni)(∑1≤i≠j≤rninj)).

3  Fuzzy Superhypergraphs

In this section, we introduce the novel concept of fuzzy supervertices, fuzzy superedges or fuzzy links, and fuzzy superhypergraph.

Example 3.1. Let X = {a, b, c, d, e, f} and consider the fuzzy hypergraph H=(X,σ1,σ2,σ3) as shown in Fig. 1. In this fuzzy hypergraph as a complex (super)hypernetwork, we can find the relation between a, b in σ1, c, d in σ2 and e, f in σ3, but we cannot find the relation between a in σ1 with to e in σ3, for instance. Also, we cannot find the impact of element c in cluster σ1 and the impact of element f in cluster σ2, for instance. In addition, in this model, we cannot find the relation or impact of cluster σi′s with σj′s, where i, j ∈ {1, 2, 3}. Let’s assume that in this real-problem, as a complex (super)hypernetwork, we want to check the influence of and the relationship all components, either component by component or component in one cluster with another cluster and cluster to cluster. By the above reason, we cannot analyze in this form, since we do not know the relation among the elements of a cluster with elements of the other cluster and a cluster to another cluster at the same time and it is the main defect. Therefore, we need this component-to-component connection and group-to-group connection in the other model as a complex (super)hypernetwork.

Figure 1: Hypergraph H=(X,σ1,σ2,σ3)

Definition 3.1. Let j,k∈N,1≤j≤k,q∈R+, H∗=(X,{Si}i=1k,{φi,j}i,j) be a quasi superhypergraph, σi={(x,σi(x))|x∈Si,0≤σi(x)≤1} and μi,j:φi,j→[0,1] be fuzzy subsets. Then H=({σi}i=1k,{μi,j}i,j) is called a q-fuzzy quasi superhypergraph on H∗, if X=⋃i=1ksupp(σi) and μi,j(x,φi,j(x))≤σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)), where w(σt)=∑xt∈Stσ(xt). We will call σi’s by fuzzy supervertices and μi,j’s by fuzzy superedges or fuzzy links from σi’s to σj’s and 1-fuzzy quasi superhypergraph is denoted by fuzzy quasi superhypergraph.

From now on, if there is not any fuzzy links from σi’s to σj’s, it means that μi,j≡0. In this case H∗=(X,{Si}i=1k,{φi,j}i,j) converts to a fuzzy hypergraph.

Example 3.2. Let X={xi}i=17. Then H=(X,{Si}i=13,{φ1,2,φ2,3,φ1,3}) is a quasi superhypergraph in Fig. 2, where φ1,2={(x1,x4),(x2,x4),(x3,x5)}, φ2,3={(x4,x6),(x5,x7)} and φ1,3={(x1,x6),(x2,x6),(x3,x7)}. Hence, H=(X,{σi}i=13,μ1,2,μ2,3,μ1,3) is a 2033-fuzzy quasi superhypergraph as Fig. 2, while is not a fuzzy quasi superhypergraph.

Figure 2: 2033-Fuzzy quasi superhypergraph H=(X,σ1,σ2,σ3,μ1,2,μ2,3,μ1,3)

Theorem 3.1. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗ and q,q′∈R+. If q′ ≤ q, then H=({σi}i=1k,{μi,j}i,j) is a q′-fuzzy quasi superhypergraph on H∗.

Proof. Let q<q′∈R+ and x ∈ X. Since H=({σi}i=1k,{μi,j}i,j) is a q-fuzzy quasi superhypergraph on H∗, come to conclusion μi,j(x,φi,j(x))≤σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj))≤σi(x)q′(w(σi))∧σj(φi,j(x))q′(w(σj)). Thus H=({σi}i=1k,{μi,j}i,j) is a q′-fuzzy quasi superhypergraph on H∗.

Proposition 3.1. The following hold:

(i)   Every fuzzy hypergraph is a fuzzy quasi superhypergraph.

(ii)   Every fuzzy graph is a fuzzy quasi superhypergraph.

Proof. The proof is clear by definition.

Theorem 3.2. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗. If for all 1≤i≤k,|Si|=1, then

(i)   For all x ∈ X, s=sup{μi,j(x,φi,j(x))|x∈X,i,j}≤1q is obtained.

(ii)   If q ∈ [t, s], then H=({σi}i=1k,{μi,j}i,j) is a fuzzy graph, where t=max{1σi(x)∧σj(φi,j(x))|x∈X,i,j∈N}.

(iii)   If q ≤ 1, then for all 1≤i≤k,|Si|=1. Then for any fuzzy subsets σi and μi,j, H=({σi}i=1k,{μi,j}i,j) is a fuzzy quasi superhypergraph on H∗.

Proof. Let x ∈ X. Then

(i) H=({σi}i=1k,{μi,j}i,j) is a q-fuzzy quasi superhypergraph on H∗ and for all 1≤i≤k,|Si|=1, so

μi,j(x,φi,j(x))≤σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj))=σi(x)qσi(x)∧σj(φi,j(x))qσj(φi,j(x))=1q.

It follows that q ≤ s.

(ii) By (i), q ≤ s. Let x ∈ X. If H=({σi}i=1k,{μi,j}i,j) is a fuzzy graph, then it must be μi,j(x,φi,j(x))≤1q≤(σi(x)∧σj(φi,j(x))). Hence for any x ∈ X and i, j, q≥1σi(x)∧σj(φi,j(x)) and so q ∈ [t, s].

(iii) It is obtained from (ii), (iii) and Theorem 3.1.

Corollary 3.1. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗. If for all 1≤i≤k,q(w(σi)>1, then μi,j(x,φi,j(x))≤σi(x)∧σj(φi,j(x)).

Definition 3.2. Let q,q′∈R+,H=({σi}i=1k,{μi,j}i,j) and H′=({σi′}i=1k′,{μi,j′}i,j) be q-fuzzy quasi superhypergraph and q′-fuzzy quasi superhypergraph on quasi superhypergraphs H∗=(X,{Si}i=1k,{φi,j}i,j) and H′∗=(X′,{Si′}i=1k,{φi,j′}i,j), respectively.

(i)   A bijective mapping h:H→H′ is called an isomorphism, if for any x ∈ X, there exists a permutation π,ν on Ik={1,2,…,k} such that σi(x)=σπ(i)′(h(x)) and μi,j(x,φi,j(x))=μπ(i)ν(j)′(h(x),φπ(i)ν(j)′(h(x)). In this case will say that H=({σi}i=1k,{μi,j}i,j) and H′=({σi′}i=1k′,{μi,j′}i,j) are isomorphic and will denote it by H ≅ H′.

(ii)   H is called a self complemented fuzzy quasi superhypergraph, if H≅Hc.

Example 3.3. Let X = {x, y, z, w} and X′ = {x′, y′, z′, w′}. Then H=(X,{σi}i=12,μ1,2) and H′=(X′,{σi′}i=12,μ1,2′) are 2-fuzzy quasi superhypergraphs as subfigures 3a, 3b in Fig. 3, where φ1,2={(x,z),(y,w)} and φ2,1′={(x′,z′),(y′,w′)}. Consider h:H→H′, by h = {(x, y′), (y, x′), (z, w′), (w, z′)} and permutations π=ν=(1,2). Then H ≅ H′.

Figure 3: Isomorphic fuzzy 2-quasi superhypergraphs H, H′

Theorem 3.3. Let H=({σi}i=1k,{μi,j}i,j) and H′=({σi′}i=1k′,{μi,j′}i,j) be q-fuzzy quasi superhypergraph and q′-fuzzy quasi superhypergraph on quasi superhypergraphs H∗=(X,{Si}i=1k,{φi,j}i,j) and H′∗=(X′,{Si′}i=1k,{φi,j′}i,j), respectively. If H ≅ H′, then q = q′.

Proof. Since H ≅ H′, there exists a bijective mapping h:H→H′ is called an isomorphism, if for any x ∈ X, there exist permutations π,ν on Ik={1,2,…,k} such that σi(x)=σπ(i)′(h(x)) and μi,j(x,φi,j(x))=μπ(i)ν(j)′(h(x),φπ(i)ν(j)′(h(x)). Without loss of generality and a rearrangement, we consider an arbitrary fuzzy supervertex σm and suppose that σm={(xi,σm(xi))|1≤i≤r} in q-fuzzy quasi superhypergraph H=({σi}i=1k,{μi,j}i,j). Thus there exists a fuzzy supervertex σm′′={(xi′,σm′(xi′))|1≤i≤r′} in q′-fuzzy quasi superhypergraph H′=({σi′}i=1k′,{μi,j′}i,j) such that σm(xi)=σm′′(xi′), which m′=π(m) and h(xi)=xi′. It follows that r = r′ and for any given 1 ≤ i, i′ ≤ r, we have xiq(∑i=1rxi)=xi′q′(∑i=1rxi′). Hence ∑i=1mxiq(∑i=1rxi)=∑i=1mxi′q′(∑i=1rxi′) and so q = q′.

Let H=({σi}i=1k,{μi,j}i,j) and H′=({σi′}i=1k′,{μi,j′}i,j) be q-fuzzy quasi superhypergraph and q′-fuzzy quasi superhypergraph on quasi superhypergraphs H∗=(X,{Si}i=1k,{φi,j}i,j) and H′∗=(X′,{Si′}i=1k,{φi,j′}i,j), respectively. From now on, based on Theorem 3.3, when we say that H ≅ H′, then q = q′.

Theorem 3.4. Let H=({σi}i=1k,{μi,j}i,j) and H′=({σi′}i=1k′,{μi,j′}i,j) be isomorphic q-fuzzy quasi superhypergraphs.

(i)   For any given fuzzy supervertex σi in H, there exists a fuzzy supervertex σi′ in H′ and λ∈R, such that w(σi)w(σj)=λ.

(ii)   For any x ∈ X, there exist λ∈R and x′ ∈ X′ such that xx′=λ.

Proof. (i) Let σi={(xm,σi(xm))|1≤m≤r} be a fuzzy supervertex in H. Clearly

h(σi)={(h(xm),σi(h(xm))|1≤m≤r} is a fuzzy supervertex in H′ and using Theorem 3.3, xmw(σi)=xm′w(h(σi)). It follows that w(σi)w(h(σi))=xmxm′.

(ii) Let x ∈ X. Since X=⋃i=1ksupp(σi), there exists a fuzzy supervertex σi in H such that x∈supp(σi). Hence by item (i), there exists a fuzzy supervertex σi′ in H′ and λ∈R, such that x∈supp(σi′) and w(σi)w(σi′)=xx′.

Example 3.4. Consider 2-fuzzy quasi superhypergraphs H=(X,{σi}i=12,μ1,2) and H′=(X′,{σi′}i=12,μ1,2′) as shown in subfigures 3a, 3b of Fig. 3. Clearly H ≅ H′, while for any i∈{1,2},w(σi)≠w(σi′). This shows that if q-fuzzy quasi superhypergraphs are isomorphic, it is not necessary that their fuzzy superverties have the same weights.

Definition 3.3. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗. Then Hc=({σic}i=1k,{μi,jc}i,j) is called a complement q-fuzzy quasi superhypergraph of H=({σi}i=1k,{μi,j}i,j), if σic=σi and μi,jc(x,φi,j(x))=(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−μi,j(x,φi,j(x)).

Lemma 3.1. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗. Then μi,jc=μi,j if and only if μi,j(x,φi,j(x))=12(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj))), for all x ∈ X and q≥12.

Theorem 3.5. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗. Then (Hc)c≅H.

Proof. Let x ∈ X. Then for any 1≤i≤k,(σic)c(x)=σic(x)=σi(x) and

(μi,jc)c(x,φi,j(x))=(σic(x)q(w(σi))∧σjc(φi,j(x))q(w(σj)))−μi,jc(x,φi,j(x))=(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−μi,jc(x,φi,j(x))=(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−((σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−μi,j(x,φi,j(x)))=μi,j(x,φi,j(x))).

It follows that for any 1≤i≤k,(σic)c=σi and (μi,jc)c=μ. Hence (Hc)c≅H.

Theorem 3.6. Let H=({σi}i=1k,{μi,j}i,j) and H′=({σi′}i=1k′,{μi,j′}i,j) be isomorphic q-fuzzy quasi superhypergraphs. Then Hc and H′c are isomorphic q-fuzzy quasi superhypergraphs and conversely.

Proof. Since H ≅ H′, there exists a bijective mapping h:H→H′ is called an isomorphism, if for any x ∈ X, there exists a permutations π,ν on Ik={1,2,…,k} such that σi(x)=σπ(i)′(h(x)) and μi,j(x,φi,j(x))=μπ(i)ν(j)′(h(x),φπ(i)ν(j)′(h(x)). Using the definition of complement, we have

μi,jc(x,φi,j(x))=(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−μi,j(x,φi,j(x))=(σπ(i)′(h(x))q(w(σπ(i)′))∧σπ(j)′(φi,j(h(x)))q(w(σπ(j)′)))−μπ(i)ν(j)′(h(x),φπ(i)ν(j)′(h(x))=μπ(i),ν(j)c(h(x),φπ(i),ν(j)(h(x)))

Hence Hc≅H′c. Conversely, let Hc≅H′c, then by Theorem 3.5, H≅(Hc)c≅(H′c)c≅H′.

Theorem 3.7. Let H=({σi}i=1k,{μi,j}i,j) be a self complemented q-fuzzy quasi superhypergraph on H∗. Then 12(∑(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))=∑μi,j(x,φi,j(x)).

Proof. Since H=({σi}i=1k,{μi,j}i,j) is a self complemented q-fuzzy quasi superhypergraph, there exists a map h:X→X such that for any x ∈ X, σi(x)=σi′(h(x)) and μi,jc(h(x),φi,j(h(x))=μi,j(x,φi,j(x)). Then (σi(h(x))q(w(σi))∧σj(φi,j(h(x))q(w(σj)))−μi,j(h(x),φi,j(h((x))=μi,j(x,φi,j(x)). Since h:X→X is a bijection, we get to the result that

∑((σi(h(x))q(w(σi))∧σj(φi,j(h(x))q(w(σj)))−μi,j(h(x),φi,j(h((x)))=∑μi,j(x,φi,j(x)),

so

∑((σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−∑(μi,j(x,φi,j(x))=∑μi,j(x,φi,j(x)).

It follows that 12(∑(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))=∑μi,j(x,φi,j(x).

Example 3.5. Let X={xi}i=16. Then H∗=(X,{Si}i=13,{φ1,2,φ2,3}) is a 4-quasi superhypergraph in Fig. 4, where φ1,2={(x1,x3),(x2,x4)} and φ2,3={(x3,x6),(x4,x7)}. If H=(X,{σi}i=13,μ1,2,μ2,3) is a fuzzy quasi superhypergraph as shown in Fig. 4, where 0<α1≤α2≤α3≤α4≤1. The equation 12(∑(σi(x)4(w(σi))∧σj(φi,j(x))4(w(σj))))=∑μi,j(x,φi,j(x)), implies that

Figure 4: Fuzzy quasi superhypergraph H=(X,σ1,σ2,σ3,μ1,2,μ2,3,μ1,3)

12(α14(α1+α2)∧α34(α3+α4)+α24(α1+α2)∧α44(α3+α4)+α34(α3+α4)∧α54(α5+α6)+α44(α3+α4)∧α64(α5+α6))=12(α14(α1+α2)+α24(α1+α2)+α34(α3+α4)+α44(α3+α4))=18=(β1+β2+β3+β4),

where β1≤α14(α1+α2),β2≤α24(α1+α2),β3≤α34(α3+α4) and β4≤α44(α3+α4). Suppose that

α1=0.1,α2=0.2,α3=0.3,α4=0.4,α5=0.5,α6=0.6,β1=0.01,β2=0.03,β3=0.04andβ4=0.17.

It follows that 12(∑(σi(x)4(w(σi))∧σj(φi,j(x))4(w(σj))))=∑μi,j(x,φi,j(x)), while H=({σi}i=1k,{μi,j}i,j) is not a self complemented fuzzy quasi superhypergraph on H∗ as shown in Fig. 5. Thus the converse of Theorem 3.7, is not necessarily true.

Figure 5: Fuzzy quasi superhypergraph Hc=(X,σ1,σ2,σ3,μ1,2,μ2,3,μ1,3) of Fig. 4

Theorem 3.8. Let H=({σi}i=1k,{μi,j}i,j) be a q-fuzzy quasi superhypergraph on H∗. If for all x ∈ X, μi,jc(x,φi,j(x))=12(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj))), then H=({σi}i=1k,{μi,j}i,j) is a self complemented fuzzy quasi superhypergraph.

Proof. Define h:X→X by h(x) = x, where x ∈ X. Clearly σc(h(x))=σc(x)=σ(x). In addition,

μi,jc(x,φi,j(x))=μi,jc(h(x),φi,j(h(x))=(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−μi,j(x,φi,j(x))=(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))−12(σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)))=12(σi(x)q(w(σi)∧σj(φi,j(x))q(w(σj)))=μi,j(x,φi,j(x)).

It follows that μi,jc(x,φi,j(x))=μi,j(x,φi,j(x)) and so H=({σi}i=1k,{μi,j}i,j) is a self complemented fuzzy quasi superhypergraph.

Definition 3.4. Let H∗=(X,{Si}i=1k,{φi,j}i,j) be a quasi superhypergraph, σi={(x,σi(x))|x∈Si,0≤σi(x)≤1} and μi,j:φi,j→[0,1] be fuzzy subsets. Then H=({σi}i=1k,{μi,j}i,j) is called a strong q-fuzzy quasi superhypergraph on H∗, if X=⋃i=1ksupp(σi) and μi,j(x,φi,j(x))=σi(x)q(w(σi))∧σj(φi,j(x))q(w(σj)), where w(σt)=∑xt∈Stσ(xt).