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ABSTRACT
This paper, applies the concept of KM-fuzzy metric spaces and introduces a novel concept of KM-fuzzy metric hypergraphs based on KM-fuzzy metric spaces. In special cases, we add some conditions to axioms of KM-fuzzy metric hypergraphs(to obtain of elementary hypergraphs, C-accessible hypergraphs, Cor-able hypergraphs, fuzzy hypergraphs) and so obtain locally strong KM-fuzzy metric hypergraphs and strong KM-fuzzy metric hypergraphs. This study, investigates on the finite KM-fuzzy metric spaces with respect to metrics, KM-fuzzy metrics and constructs KM-fuzzy metric spaces on any given non-empty sets. It tries to extend the concept of KM-fuzzy metric spaces to union of KM-fuzzy metric spaces and product of KM-fuzzy metric spaces and in this regard investigates on union and product of KM-fuzzy metric hypergraphs.
1. Introduction
The mathematical concept of theory of hypergraphs has been introduced by Berge as a generalisation of theory of graph with the motivation that hypergraphs are an extension of classical results of graph theory around 1960 [Citation1]. One of the applications of hypergraph structures is a modelling for the complex hypernetworks. Since sometimes graph structures give very limited information about of complex networks, so we say that the main motivation of hypergraph structures is for covering graph defects in the applications. Also the notion of hypergraph has been considered as a basic tool to present the hyperstructure of a system by the clustering and partitating methods. Today, hypergraphs have important applications and are used in complex hypernetworks such as computer science, wireless sensor hypernetwork, machine learning. So there has been a lot of researches about using hypergraphs in relational databases, which might be viewed as a sort of data mining. There are also some researches about networks where matroids and hypergraphs are used together in the demonstrations [Citation2]. In classical set theory, the mathematical concepts introduce purely and without any quality or criteria, that it is not attractive to use in world. So Zadeh introduced the concept of fuzzy set theory as one of a generalisation of set theory to deal with uncertainties [Citation3]. This theory describes an important role in modelling and controlling unsure hypersystems in nature, society and industry. Follow this, fuzzy topological spaces are a generalisation of topological spaces and have a fundamental role in construction of fuzzy metric spaces as a extension of the concept of metric spaces. Theory of fuzzy metric space works on distance between of two points as nonnegative fuzzy numbers and has some applications. The structure of fuzzy metric space, is equipped with mathematical tools such as triangular norms and fuzzy subsets dependent on time parameter and other variables. This theory has been proposed by different researchers with different definitions points view [Citation4–7], that this study applies a notion of KM-fuzzy metric space by Kramosil and Michalek introduced in 1975 [Citation6]. Kaufmann [Citation8], introduced and provided the theory of fuzzy hypergraphs as a generalisation of concept of hypergraphs, in such a way that fuzzy hypergraphs have important applications to decision making, mobile network and similar applications [Citation9,Citation10]. Further materials regarding graphs and hypergraphs are available in the literature too [Citation11–23].
Regarding these points, we introduce the novel concept of KM-fuzzy metric hypergraphs. Also with respect to the concept of KM-fuzzy metric hypergraphs we proved that for every given set one can construct a metric space. KM-fuzzy metric spaces have some important tools as triangular norms and fuzzy subsets, so it is one of main reasons for choosing the KM-fuzzy metric spaces to construct of the KM-fuzzy metric hypergraphs. Although the parameter of time(unlimited time) in KM-fuzzy metric spaces is a problem in our study, but the concept of fuzzy hypergraphs and their applications lead us to introduce this concept. We apply the KM-fuzzy metrics to measurement limitation on vertices and hyperedges of KM-fuzzy metric hypergraphs, so the KM-fuzzy metrics play a key role in our study. The fuzzy metric spaces are not necessarily finite space, so one of our motivation of this work is a construction of finite KM-fuzzy metric space and their extension based on KM-fuzzy metric hypergraphs. The main of our motivation of this work is to present the concept of fuzzy hypergraphs based on t-norm such as Domby t-norm, Godel t-norm and etc. We applied the notation of KM-fuzzy metric space to generate of finite KM-fuzzy metric hypergraph. It is extended some production operations on KM-fuzzy metric spaces and so it is generalised the KM-fuzzy metric hypergraphs to larger class. Also the notation of fundamental sequence in KM-fuzzy metric hypergraphs is introduced and based on fundamental sequence the concept of core hypergraphs is presented.
2. Preliminaries
In this section, we recall some definitions and results, which we need in what follows.
Definition 2.1
[Citation1,Citation15]
Let X be a finite set. A hypergraph on X is a pair such that for all
and
. The elements
of X are called
hyper
vertices, and the sets
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
is called a complete hypergraph, if for any
there is
such that
. A hypergraph
is called as a joint complete hypergraph, if
for all
and
element
loop
. If for all
, the hypergraph becomes an ordinary
undirected
graph. and n rows representing the vertices
, where for all
and for all
we have
if
and
if
.
Definition 2.2
[Citation24]
Let X be a finite set and be a finite family of non trivial fuzzy subsets of X such that
, where
. Then the pair
is called a fuzzy hypergraph on X and
is called the fuzzy edge set of
which is denoted by
. The height of
is defined by
, where for all
denotes the height of μ.
Definition 2.3
[Citation25]
A binary operation is a t-norm if it for all
satisfies the following:
;
;
;
If
and
then
.
Definition 2.4
[Citation26]
A triplet is called a KM-fuzzy metric space, if X is an arbitrary non–empty set, T is a left-continuous t-norm and
is a fuzzy set, such that for each
and
, we have:
,
for all t>0,
(commutative property),
(triangular inequality),
is a left-continuous map,
,
implies that x = y.
If satisfies in conditions (i)–(vii), then it is called KM-fuzzy pseudometric space and ρ is called a KM-fuzzy pseudometric. a fuzzy version of the triangular inequality. The value
is considered as the degree of nearness from
Theorem 2.5
[Citation26]
Let be a KM-fuzzy metric space. Then
is a non-decreasing map.
3. Extension Operations on KM-Fuzzy Metric Spaces
In this section, we extend the concept of KM-fuzzy metric spaces to union and product of KM-fuzzy metric spaces. From now on, for all we consider
,
,
Let
and
be KM-fuzzy metric spaces,
and
. For an arbitrary
, define
by
. So we have the following theorem.
Theorem 3.1
Let and
be KM-fuzzy metric spaces. Then
is a KM-fuzzy metric space.
Proof.
Let and
.
Since for all
and
, we have
.
if and only if
if and only if
if and only if
.
It is clear that
is a commutative map.
Since
are left-continuous maps, we get that ρ is a left continuous map.
Let
. Then
,
. Thus
is a KM-fuzzy metric space.
Let
and
be KM-fuzzy metric spaces,
and
. Consider
, define
by
So we have the following theorem.
Theorem 3.2
Let and
be KM-fuzzy metric spaces. Then
is a KM-fuzzy metric space, where
.
Proof.
Let and
. We only prove the triangular inequality property and other cases are immediate.
Let (for
is similar), then
If
, then
If
, then
Let . Then
If
, since
and
, we get that
and so
If
, since
and
, we get that
and so
It follows that
is a KM-fuzzy metric space.
Corollary 3.3
Let and
be KM-fuzzy metric spaces, where
. Then
is a KM-fuzzy metric space.
Theorem 3.4
Let be a KM-fuzzy metric space and φ be a bijection on X. Then there exists a fuzzy set
such that
is a KM-fuzzy metric space.
Proof.
Let x, y, X and . Define
by
. It is clear that
is a KM-fuzzy metric space.
4. KM-Fuzzy Metric Hypergraph
In this section, we introduce a novel concept as ((locally)strong, trivial, elementary) KM-fuzzy metric hypergraphs and analyse some of their properties. Also, we investigate the notation of fundamental sequence and generate some type of core hypergraphs in any given KM-fuzzy metric hypergraph.
A fuzzy version of the triangular inequality. The value is considered as the degree of nearness from
Definition 4.1
Let be a KM-fuzzy metric space and
be a family of fuzzy subset of X. If there exists
for t = 0, we call starting time
such that for all
and for all
,
, then
is called a KM-fuzzy metric hypergraph on X and
is called the KM-fuzzy metric edge set of H. The incidence matrix of a KM-fuzzy metric hypergraph is a matrix
, with m columns representing the fuzzy hyperedges
and n rows representing the vertices
, where
if
,
if
(∞ is a symbol, means that
).
Remark 4.2
By Definition 2.2, in any fuzzy hypergraph , since
, for all
and
, we get that
. But in given a KM-fuzzy metric hypergraph
, may there exist
and
in such a way that
.
Proposition 4.3
Let be a KM-fuzzy metric hypergraph on X. Then for all
and for all
in starting time we have
or
.
Proof.
Let and
. Since
is a KM-fuzzy metric hypergraph on X, we get that
. Hence
and so
or
.
Example 4.4
Let and
. For all
define
and
. Thus
is a KM-fuzzy metric hypergraph with t = 1 (it is not a fuzzy hypergraph) in Figure . For simplify, we compute
in Table . Since
and
, we get
.
Theorem 4.5
Let be a KM-fuzzy metric space and
be a family of fuzzy subset of X and
.
If for all
then
is a KM-fuzzy metric hypergraph on X.
If for all
and for all
we have
then
is a KM-fuzzy metric hypergraph on X.
Table 1. The corresponding incidence matrix of Figure .
Proof.
(i) Let . Since for all
, we get that there exists
such that
and for all
we have
. Let
be an arbitrary and fixed. Then for all
and
. It follows that
is a KM-fuzzy metric hypergraph on X.
(ii) Let and for all
and for all
clearly
. Because for all
and for all
we have
, so
implies
. Thus
is a KM-fuzzy metric hypergraph on X.
A KM-fuzzy metric hypergraph which is satisfied in condition of Theorem 4.5, will call an elementary KM-fuzzy metric hypergraph.
Definition 4.6
Let be a KM-fuzzy metric space and
be a family of fuzzy subset of X. If there exists
(for t = 0, we call starting time) such that for all
and for all
, if
, then
is called a strong KM-fuzzy metric hypergraph on X and
is called the KM-fuzzy metric edge set of H. In strong KM-fuzzy metric hypergraph
, for all
if
, then x can be equal to y, so
. If there exists
and
in such a way that
, we say that
is a locally strong KM-fuzzy metric hypergraph.
Theorem 4.7
Let X be a KM-fuzzy metric space, be a family of fuzzy subsets of X and
. Then there exists a KM-fuzzy metric ρ on X such that
is a strong KM-fuzzy metric hypergraph on X.
Proof.
Let and
. Define
and
One can see that in non starting time,
is a strong KM-fuzzy metric hypergraph on X.
In the following, for given any non-empty set one can construct a KM-fuzzy metric space and so a KM-fuzzy metric hypergraph.
Corollary 4.8
Let X be a non-empty set, be a family of fuzzy subset of X and
. Then there exists a fuzzy subset
such that
is a KM-fuzzy metric space and
is a KM-fuzzy metric hypergraph on X.
Definition 4.9
Let be a KM-fuzzy metric hypergraph. If for all
, then
is called a trivial KM-fuzzy metric hypergraph.
Corollary 4.10
Let be a trivial KM-fuzzy metric hypergraph and
be the incidence matrix of H. Then for all
and
we have
or
.
Let be a KM-fuzzy metric hypergraph on X and
, where
. Consider
and
, where
.
In the following, we extract hypergraphs from KM-fuzzy metric hypergraphs and α-level subsets.
Theorem 4.11
Let in non starting time, be a KM-fuzzy metric space and
be a KM-fuzzy metric hypergraph on X.
For all
is a hypergraph.
If
is a trivial KM-fuzzy metric hypergraph, then for all
is a hypergraph.
If
is a strong KM-fuzzy metric hypergraph, then for all
is a hypergraph.
If for all
,
, and
then for all
is a hypergraph.
If
is a hypergraph and
then
.
Proof.
(i) We show that . Since
is a non-trivial KM-fuzzy metric hypergraph, there exists
such that
. Then for all
,
and so
.
(ii) Since for all ,
, it is seen that
. Thus
and so
is a hypergraph.
(iii) Since in non starting time, is a strong KM-fuzzy metric hypergraph, for all
and
, we obtain that
. Hence
implies that
and
. It follows that for all
,
and so
is a hypergraph.
(iv) Let . Then for all
we conclude that
. Since for all
, we get that
.
. So we get
such that
. It follows that for all
is a hypergraph.
(v) Let . Then there is
such that
and so
. It follows that
and so
.
Theorem 4.12
Let be a KM-fuzzy metric hypergraph. Then
if
then
is a fuzzy hypergraph,
is a loop
joint
hypergraph, where
is height of
.
Proof.
(i) It is clear by definition.
(ii) Since , for all
we get that
and so
is a loop (joint) hypergraph.
Theorem 4.13
Let be a strong KM-fuzzy metric hypergraph. Then in non starting time
is a fuzzy hypergraph.
Proof.
Since is a strong KM-fuzzy metric hypergraph, we get that for all
,
. Because t is not in starting time, then
and so
. It follows that for all
,
and so
is a fuzzy hypergraph.
In what follows, we can generate KM-fuzzy metric hypergraphs from fuzzy hypergraphs.
Theorem 4.14
Let be a fuzzy hypergraph. Then there exists a fuzzy subset
such that in each non starting time,
is a KM-fuzzy metric hypergraph.
Proof.
Let . Since
is a fuzzy hypergraph, there exists
in such a way that
. For all
, define
and
where
. One can see that each non starting time,
is a KM-fuzzy metric hypergraph.
Corollary 4.15
Let X be a non-empty set and . Then there exists a fuzzy subset
such that
is a KM-fuzzy metric space.
Proof.
Firstly we construct a fuzzy hypergraph . Then by Theorem 4.14, there is a fuzzy subset
such that
is a KM-fuzzy metric space.
Theorem 4.16
Let in non starting time, be an elementary KM-fuzzy metric hypergraph. Then
is an elementary hypergraph.
Proof.
Let . Then
and so
. By Theorem 4.11,
is elementary hypergraph.
Definition 4.17
Let be a KM-fuzzy metric hypergraph and
. The sequence of real numbers
is called a fundamental sequence, if (i),
, implies that
and (ii),
. Also
and
is called the set of core hypergraphs H.
Example 4.18
Let and
. For all
define
and
. Thus for t = 1,
is a KM-fuzzy metric hypergraph in Table .
Table 2. KM-fuzzy metric hypergraph .
Definition 4.19
Let be a KM-fuzzy metric hypergraph and C be a class of hypergraphs on X. Then
is a C-accessible hypergraph or say C is a
-derivable or
-able hypergraph, if
.
Theorem 4.20
Let be a KM-fuzzy metric hypergraph with incidence matrix
and
. If for all
and
then
Proof.
(i, ii) Let . Because for all
and
, there exists
in such a way that
. But
is a chain, so we can rearrange
In addition for all
and for all
we get that
. It is clear that
Theorem 4.21
Let be a strong elementary KM-fuzzy metric hypergraph. Then
Proof.
(i, ii, iii) Since is a strong elementary KM-fuzzy metric hypergraph, for all
and
so by Theorem 4.20, the proof is completed.
Corollary 4.22
Let X be a non-empty set. Then any elementary hypergraph on X is a -able hypergraph.
Theorem 4.23
Let X be a non-empty set. Then any joint complete hypergraph on X is a -able hypergraph.
Proof.
Let and
. By Corollary 4.8, define a fuzzy hypergraph
by incidence matrix
, where
, if
and
, if i<j. Thus
and so
. It follows that
, where C is class of joint complete hypergraphs on X.
A hypergraph is called a discrete complete hypergraph, if for any
and
. Let
be a discrete complete hypergraph,
and
Denote the set of all discrete complete hypergraphs with
on H (m–hyperedges), by
and the set of all discrete complete hypergraphs on H, by
.
Theorem 4.24
Let X be a non-empty set. Then any discrete complete hypergraph on X is a -able hypergraph.
Proof.
Let and
. By Corollary 4.8, define a fuzzy hypergraph
by incidence matrix
, where
,
and
, for other cases. Thus
and so
,
. It follows that
, where
is class of discrete complete hypergraphs on X.
5. Operation on KM-Fuzzy Metric Hypergraphs
F. Harary [Citation27] and L. Maninska [Citation28], introduced and presented some of type of product on hypergraphs and get new hypergraphs. In this section, we will define some products on KM-fuzzy metric hypergraphs and via these productions, get new KM-fuzzy metric hypergraphs. From now on, we consider ,
as KM-fuzzy metric hypergraphs on sets
and
, respectively.
Remember that for fuzzy subsets μ and on X and
, square product of fuzzy subset
by
[Citation24].
Definition 5.1
Let ,
be KM-fuzzy metric hypergraphs on sets
and
, respectively. Define square product of KM-fuzzy metric hypergraphs by
, where
and
.
Example 5.2
Let and
. For all
define
,
, for all
and
. Thus
and
are KM-fuzzy metric hypergraphs for
and
, respectively as Figure .
Thus we obtain the square product of KM-fuzzy metric hypergraph in Figure .
For instance, we have
Theorem 5.3
Let and
be KM-fuzzy metric hypergraphs on
respectively. Then
For all
and
we have
.
is a KM-fuzzy metric hypergraph on
.
Proof.
(i) Let . Then
, so
and
. Hence
and
and so
. The conversely is similar to.
(ii) For all and for all
, since
is a KM-fuzzy metric hypergraphs on
and
is a KM-fuzzy metric hypergraphs on
, for some
, take
so by Theorem 2.5, we get that
Thus
is a KM-fuzzy metric hypergraph on
.
Definition 5.4
Let ,
be KM-fuzzy metric hypergraphs on sets
and
, respectively. Define Cartesian product of KM-fuzzy metric hypergraphs by
, where
and the Cartesian product of fuzzy subsets
are defined by
. Indeed,
such that for all
and
(i) for and
or
(ii) for and
.
Example 5.5
Consider the fuzzy hypergraphs in Example 5.2. Thus we obtain the square product of KM-fuzzy metric hypergraph for t = 2, in Table and Figure . For instance, we compute
and
. We have
. By definition
Table 3. KM-fuzzy metric hypergraph for t = 2.
Theorem 5.6
Let ,
be KM-fuzzy metric hypergraphs on
respectively. Then
For all
and
we have
.
is a KM-fuzzy metric hypergraph on
.
Proof.
(i) Let . Then
, so
and
. Hence
and
and so
. The conversely is similar to.
(ii) Let such that for all
and
for
and
or
for
and
. Let
and
. Then
and
or
and
. If
and
, then
. If
and
, then
For all
and for all
, since
is a KM-fuzzy metric hypergraphs on
and
is a KM-fuzzy metric hypergraphs on
, for some
, take
so by Theorem 2.5, we get that
Thus
is a KM-fuzzy metric hypergraph on
.
Definition 5.7
Let ,
be KM-fuzzy metric hypergraphs on sets
and
, respectively. Define the minimal rank preserving direct product of fuzzy subset
by
such that for all
and for
and
is a multiset of elements of
such that
and define minimal rank preserving direct product of KM-fuzzy metric hypergraphs by
, where
.
Example 5.8
Consider the fuzzy hypergraphs in Example 5.2. Thus we obtain the minimal rank preserving direct product of KM-fuzzy metric hypergraph for t = 2, in Table and Figure .
Table 4. KM-fuzzy metric hypergraph ) for t = 2.
For instance, gives some following hyperedges:
Theorem 5.9
Let and
be KM-fuzzy metric hypergraphs on
respectively. Then
For all
and
, we have
.
is a KM-fuzzy metric hypergraph on
.
Proof.
(i) Let . Then
, so
and
. Hence
and
and so
. The conversely is similar to.
(ii) We have such that for all
and for
and
is a multiset of elements of
such that
. For all
and for all
, since
is a KM-fuzzy metric hypergraphs on
and
is a KM-fuzzy metric hypergraphs on
, for some
, take
so by Theorem 2.5, we get that
Thus
is a KM-fuzzy metric hypergraph on
.
Definition 5.10
Let ,
be KM-fuzzy metric hypergraphs on sets
and
, respectively. Define the maximal rank preserving direct product of fuzzy subset
by
such that for all
and
(i) for and
is a multiset of elements of
such that
or
(ii) for and
is a multiset of elements of
such that
and define maximal rank preserving direct product of KM-fuzzy metric hypergraphs by
, where
.
Theorem 5.11
Let and
be KM-fuzzy metric hypergraphs on
respectively. Then
For all
and
we have
.
is a KM-fuzzy metric hypergraph on
.
Proof.
It is similar to Theorem 5.9.
Definition 5.12
Let ,
be KM-fuzzy metric hypergraphs on sets
and
, respectively. Define the strong minimal product of fuzzy subset
by
if and only if
So define strong minimal product of KM-fuzzy metric hypergraphs by
, where
.
Example 5.13
Consider the fuzzy hypergraphs in Example 5.2. Thus we obtain the strong minimal product of KM-fuzzy metric hypergraph, for t = 2 in Table . For instance, we compute
. We have
. By definition
Table 5. KM-fuzzy metric hypergraph for t = 2.
Theorem 5.14
Let and
be KM-fuzzy metric hypergraphs on
respectively. Then
For all
and
we have
.
is a KM-fuzzy metric hypergraph on
.
Proof.
It is obtained by Theorem 5.6 and 5.9.
Definition 5.15
Let ,
be KM-fuzzy metric hypergraphs on sets
and
, respectively. Define the strong maximal product of fuzzy subset
by
if and only if
So define strong product of KM-fuzzy metric hypergraphs by
, where
.
Theorem 5.16
Let and
be KM-fuzzy metric hypergraphs on
respectively. Then
For all
and
we have
.
is a KM-fuzzy metric hypergraph on
.
Proof.
It is similar to Theorem 5.14.
Definition 5.17
Let be a KM-fuzzy metric space. If
is a KM-fuzzy metric hypergraph on X, then for any
, and each
that
, define the complement of fuzzy subset
, where
and
.
We will denote the complement of a KM-fuzzy metric hypergraph , by
, where
.
Example 5.18
Consider the KM-fuzzy metric hypergraph in Example 5.2. Thus we obtain the complement of KM-fuzzy metric hypergraph, for t = 1 in Figure .
For instance,
. In similar a way,
.
Theorem 5.19
Let be a KM-fuzzy metric space and
be a KM-fuzzy metric hypergraph on X. Then
for all
and
we have
if
then for all
we have
where
for all
and
we have
.
Proof.
(i) Let and
. Then
(ii) Let and
. Since
, we get that
.
(iii) Let and
. Then by item (i), we get that
.
Theorem 5.20
Let be a KM-fuzzy metric space,
be a KM-fuzzy metric hypergraph on X,
and
. Then
if
then
if
then
implies that
if
then
implies that
if
then
implies that
Proof.
Obviously it is obtained.
Let be a KM-fuzzy metric space,
be a KM-fuzzy metric hypergraph on X. For simplify, we denote
and for all
Corollary 5.21
Let be a KM-fuzzy metric space,
be a KM-fuzzy metric hypergraph on X,
and
. Then
if
then
implies that
if
then
.
Theorem 5.22
Let be a KM-fuzzy metric space and
be a KM-fuzzy metric hypergraph on X. Then
for all
and
we have
is a KM-fuzzy metric hypergraph.
Proof.
(i) Let and
. Since
, by definition we have
. In a similar way one can see that
and so
.
(ii) Let and
. Then by Theorem 5.19,
Since
, for some
we get that
It follows that
is a KM-fuzzy metric hypergraph.
Theorem 5.23
Let be a KM-fuzzy metric space,
be a KM-fuzzy metric hypergraph on X and
. Then
implies that
implies that
if
, then
implies that H is a locally strong KM-fuzzy metric hypergraph.
Proof.
(i, ii) Since , we get that for all
. It follows that for all
where
and
. Thus for all
we have
. It concludes that
and so
. Hence x = y and
and so
.
(iii, iv) Since , we get that for all
. It follows that there exists
where
and
. Thus there exists
in such a way that
. It concludes that
and so
.
Corollary 5.24
Let be a KM-fuzzy metric space,
is a KM-fuzzy metric hypergraph on X and
. Then
if and only if
if
then for all
we have
.
6. Discussion of Results and Conclusion
The current paper has introduced a novel concept of fuzzy algebra as KM-fuzzy metric hypergraph. Indeed it has presented a new generalisation of hypergraphs based on KM-fuzzy metric spaces. This work extended and obtained some properties in KM-fuzzy metric spaces. Based on KM-fuzzy metric hypergraphs, every non empty set converted to a KM-fuzzy metric space. It is showed that the product and union of KM-fuzzy metric spaces is a KM-fuzzy metric space, the extended KM-fuzzy metric spaces are constructed using of the some algebraic operations on KM-fuzzy metric spaces. Moreover, the concept of complement of KM-fuzzy metric hypergraphs is defined and investigated some its properties. Incidence matrix of KM-fuzzy metric hypergraphs, is introduced and investigated some their properties. Using the concept of valued-cuts, we connected the concept of KM-fuzzy metric hypergraphs to hypergraphs. In addition, fundamental sequence as a tool in KM-fuzzy metric hypergraphs is introduced and based on fundamental sequence the concept of core hypergraphs is presented. In study of KM-fuzzy metric hypergraphs, despite having key mathematical tools there are some limitations. The union of two KM-fuzzy metric hypergraphs is not necessarily, a KM-fuzzy metric hypergraph so the class of KM-fuzzy metric hypergraphs is not closed under any given algebraic operation. In addition KM-fuzzy metric hypergraphs are different with fuzzy hypergraphs, so could not generalise the capabilities of fuzzy hypergraphs to KM-fuzzy metric hypergraphs.
We hope that these results are helpful for further studies in theory of graphs. In our future studies, we hope to obtain more results regarding instuitic metric graphs, neutrosophic metric graphs, KM-neutrosophic metric hypergraphs, bipolar KM-fuzzy metric graphs, automorphism of KM-fuzzy metric graphs and reducing complexity with respect to KM-fuzzy metric directed hypergraphs.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Notes on contributors
Mohammad Hamidi
Mohammad Hamidi was born in Vidouja Village, Kashan, Isfahan, Iran in 1979. He received the B.S. degrees in mathematics from the University of Kashan, Kashan, in 2002, the M.S. degrees in mathematics (algebra) from the Isfahan University of Technology, Isfahan in 2005 and the Ph.D. degree in in mathematics (algebras and hyperalgebras and fuzzy logic) from Tehran Payame Noor University, Tehran, in 2014. From 2002 to 2005, he was a Research Assistant with the Mathematics University. Since 2014, he has been an Assistant Professor with the Mathematics Department, Payame Noor University of Tehran. Since 2019, he has been an Associate Professor with the Mathematics Department, Payame Noor University of Tehran. He is the author of two books, more than 70 articles, and more than 3 research projecst. His research interests include applications of fuzzy graph and single valued veutrosophic graphs in hypernetworks and complex hypernetworks such as Wireless sensor networks.
Sirus Jahanpanah
Sirus Jahanpanah was born in bavaryan Village, roozabad, fars, Iran in 1977. He received the B.S. degrees in mathematics from the University of Shiraz, Shiraz, in 2002, the M.S. degrees in mathematics (algebra) from the University of shiraz, shiraz in 2006 and the Ph.D. student in mathematics (algebras and hyperalgebras and fuzzy logic) at the Tehran Payame Noor University, Tehran . Since 2007, he has been an Instructor with the Mathematics Department, Payame Noor University of Fars. He is the author of two books, more than 7 articles, and more than 4 research projecst.
Akefe Radfar
Akefe Radfar was born in Isfahan, Iran in 1977. She received the B.S. degrees in mathematics from the University of Isfahan, Isfahan, in 1999, the M.S. degrees in mathematics (algebra) from the Shahrekord University of Shahrekord, in 2005 and the Ph.D. degree in mathematics (algebras and hyperalgebras and fuzzy logic) from Tehran Payame Noor University, Tehran, in 2011. From 2008 to 2010, she was a Research Assistant with the Mathematics University. Since 2010, she has been an Assistant Professor with the Mathematics Department, Payame Noor University of Tehran. she has more than 30 articles.
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