1. Introduction
In 1934, Frederic Marty [
1] defined the concept of a hypergroup as a natural generalization of a group. It is well known that the composition of two elements in a group is an element, whereas the composition of two elements in a hypergroup is a non-empty set. The law characterizing such a structure is called the multi-valued operation, or hyperoperation and the theory of the algebraic structures endowed with at least one multi-valued operation is known as the Hyperstructure Theory. Marty’s motivation to introduce hypergroups is that the quotient of a group modulo of any subgroup (not necessarily normal) is a hypergroup. Significant progress in the theory of hyperstructures has been made since the 1970s, when its research area was enlarged and different hyperstructures were introduced (e.g., hyperrings, hypermodules, hyperlattices, hyperfields, etc.). Many types of hyperstructures have been used in different contexts, such as automata theory, topology, cryptography and code theory, geometry, graphs and hypergraphs, analysis of the convex systems, finite groups’ character theory, theory of fuzzy and rough sets, probability theory, ethnology, and economy [
2]. An overview of the most important works and results in the field of hyperstructures up to 1993 is given in the book by Corsini [
3], this book was followed in 1994 by the book by Vougiouklis [
4]. An overview regarding the applications of hyperstructure theory is given in the book by Corsini and Leoreanu [
5]. The book by Davvaz and Leoreanu-Fotea [
6] deals with the hyperring theory and applications. A more recent book [
7] gives an introduction into fuzzy algebraic hyperstructures.
A set is a well-defined collection of distinct objects, i.e., every element in a set occurs only once. A generalization of the notion of set was introduced by Yager [
8]. He introduced the bag (multiset) structure as a set-like object in which repeated elements are significant. He discussed operations on multisets, such as intersection and union, and he showed the usefulness of the new defined structure in relational databases. These new structures have many applications in mathematics and computer science [
9]. For example, the prime factorization of a positive integer is a multiset in which its elements are primes (e.g., 90 has the multiset
). Moreover, the eigenvalues of a matrix (e.g., the
lower triangular matrix
with
,
,
has the multiset
) and roots of a polynomial (e.g., the polynomial
over the field of complex numbers has the multiset
) can be considered as multisets.
As an extension of the classical notion of sets, Zadeh [
10] introduced a concept similar to that of a set but whose elements have degrees of membership and he called it fuzzy set. In classical sets, the membership function can take only two values: 0 and 1. It takes 0 if the element belongs to the set, and 1 if the element does not belong to the set. In fuzzy sets, there is a gradual assessment of the membership of elements in a set which is assigned a number between 0 and 1 (both included). Several applications for fuzzy sets appear in real life. We refer to the papers [
11,
12,
13]. Yager, in [
8], generalized the fuzzy set by introducing the concept of fuzzy multiset (fuzzy bag) and he discussed a calculus for them in [
14]. An element of a fuzzy multiset can occur more than once with possibly the same or different membership values. If every element of a fuzzy multiset can occur at most once, we go back to fuzzy sets [
15].
In [
16], Onasanya and Hoskova-Mayerova defined multi-fuzzy groups and in [
17,
18], the authors defined fuzzy multi-polygroups and fuzzy multi-
-ideals and studied their properties. Moreover, Davvaz in [
19] discussed various properties of fuzzy hypergroups. Our paper generalizes the work in [
16,
17,
19] to combine hypergroups and fuzzy multisets. More precisely, it is concerned with fuzzy multi-hypergroups and constructed as follows: After the Introduction,
Section 2 presents some preliminary definitions and results related to fuzzy multisets and hypergroups that are used throughout the paper.
Section 3 introduces, for the first time, fuzzy multi-hypergroups as a generalization of fuzzy hypergroups and studies its properties. Finally,
Section 4 defines some operations (e.g., intersection, selection, product, etc.) on fuzzy multi-hypergroups and discusses them.
3. Construction of Fuzzy Multi-Hypergroups
Inspired by the definition of the multi-fuzzy group [
25] and the fuzzy multi-polygroup [
17], we introduce the concept of the fuzzy multi-hypergroup. Further, we investigate their properties. It is well known that groups and polygroups [
26] are considered as special cases of hypergroups. Hence, the results in this section can be considered as more general than that in [
17,
25].
Definition 6 ([
7])
. Let be a hypergroup and μ be a fuzzy subset of H. Then μ is called a fuzzy subhypergroup of H if for all , the following conditions hold.- 1.
;
- 2.
for every there exists such that and ;
- 3.
for every there exists such that and .
Definition 7. Let be a hypergroup. A fuzzy multiset A over H is a fuzzy multi-hypergroup of H if for all , the following conditions hold.
- 1.
(or equivalently, for all );
- 2.
for every there exists such that and ;
- 3.
for every there exists such that and .
Remark 1. It is clear, using Definitions 6 and 7, that if is a hypergroup and μ is a fuzzy subhypergroup of H then μ is a fuzzy multi-hypergroup of H. Hence, the results of fuzzy subhypergroups are considered a special case of our work.
Remark 2. Let A be a fuzzy multiset over a commutative hypergroup . Then conditions 2. and 3. of Definition 7 are equivalent. Hence, to show that A is a fuzzy multi-hypergroup of H, it suffices to show that either conditions 1. and 2. of Definition 7 are valid or conditions 1. and 3. of Definition 7 are valid.
We present different examples on fuzzy multi-hypergroups.
Example 9. Let be the hypergroup defined by the following table:It is clear that is a fuzzy multi-hypergroup of H. Example 10. Let be the hypergroup defined by the following table:It is clear that is a fuzzy multi-hypergroup of . Example 11. Let be the hypergroup defined in Example 8. It is clear that A, with the fuzzy count function , is a fuzzy multi-hypergroup of . Where Proposition 1. Let be a hypergroup and A be a fuzzy multi-hypergroup of H. Then the following assertions are true.
- 1.
for all and ;
- 2.
for all .
Proof. Proof of 1. By mathematical induction on the value of n, for all is true for . Assume that for all and let . Then there exists such that . Having A a fuzzy multi-hypergroup implies that . And using our assumption that implies that our statement is true for .
Proof of 2. The proof follows from 1. by setting for all .
□
Example 12. Let be any hypergroup and be a fixed element. We define a fuzzy multiset A of H with fuzzy count function as for all . Then A is a fuzzy multi-hypergroup of H (the constant fuzzy multi-hypergroup).
Remark 3. Let be a hypergroup. Then we can define at least one fuzzy multi-hypergroup of H, which is mainly the one that is described in Example 12.
Proposition 2. Let be the biset hypergroup and A be a fuzzy multiset of H. Then A is a fuzzy multi-hypergroup of H.
Proof. Let A be a fuzzy multiset of H. Since is a commutative hypergroup, it suffices to show that conditions 1. and 2. of Definition 7 are satisfied. (See Remark 2). For condition 1., let and . It is clear that . For condition 2., let . Then there exists such that and . Therefore, A is a fuzzy multi-hypergroup of H. □
Proposition 3. Let be the total hypergroup and A be a fuzzy multiset of H. Then A is a fuzzy multi-hypergroup of H if and only if A is the fuzzy multiset described in Example 12.
Proof. If A is the fuzzy multiset described in Example 12 then it is clear that A is a fuzzy multi-hypergroup of H. Let A be a fuzzy multi-hypergroup of H and . For all , we have and . The latter and having A a fuzzy multi-hypergroup of H implies that and . Thus, for all . □
Notation 1. Let be a hypergroup, A be a fuzzy multiset of H and . We say that if .
Definition 8. Let be a hypergroup and A be a fuzzy multiset of H. Then .
Proposition 4. Let be a hypergroup and A be a fuzzy multi-hypergroup of H. Then is either the empty set or a subhypergroup of H.
Proof. Let . We need to show that the reproduction axiom is satisfied for . We show that and is done similarly. For all and , we have . The latter implies that and hence, . Moreover, for all , Condition 2. of Definition 7 implies that there exist such that and . The latter implies that and . Thus, . □
Definition 9. Let be a hypergroup and be fuzzy multisets of H. Then is defined by the following fuzzy count function. Theorem 1. Let be a hypergroup and A be a fuzzy multiset of H. If A is a fuzzy multi-hypergroup of H then .
Proof. Let . Then for all . The latter implies that . Thus, . Having a hypergroup and A a fuzzy multi-hypergroup of H implies that for every there exist such that and . Moreover, we have . Thus, . □
Notation 2. Let be a hypergroup, A be a fuzzy multiset of H and . We say that if and for all . If and then we say that and are not comparable.
In [
19], Davvaz studied fuzzy hypergroups and proved some important results about them related to level subhypergroups of fuzzy hypergroups. In what follows, we do suitable changes to extend the results of [
19] to fuzzy multi-hypergroups.
Theorem 2. Let be a hypergroup, A a fuzzy multiset of H with fuzzy count function and where for and . Then A is a fuzzy multi-hypergroup of H if and only if is either the empty set or a subhypergroup of H.
Proof. Let be a subhypergroup of H and . By setting , we get that . Having a subhypergroup of H implies that for all , . We prove condition 2. of Definition 7 and condition 3. is done similarly. Let and . Then . Having a subhypergroup of H implies that . The latter implies that there exist such that . Thus, .
Conversely, let A be a fuzzy multi-hypergroup of H and . We need to show that for all . We prove that and is done similarly. Let . Then for all . The latter implies that . Thus, . Let . Since A is a fuzzy multi-hypergroup of H, it follows that there exist such that and . The latter implies that and hence, . □
Proposition 5. Let be a hypergroup, A a fuzzy multiset of H with fuzzy count function and where for and , . If and then there exist no such that .
Proof. Let and suppose that there exist such that . Then and which contradicts the given. □
Proposition 6. Let be a hypergroup and S be a subhypergroup of H. Then for some where for , , and .
Proof. Let
where
for
and define the fuzzy multiset
A of
H as follows:
It is clear that
. We still need to prove that
is a fuzzy multi-hypergroup of
H. Using Theorem 2, it suffices to show that
is a subhypergroup of
H for all
with
for
. One can easily see that
Thus,
is either the empty set or a subhypergroup of
H. □
4. Operations on Fuzzy Multi-Hypergroups
In this section, we define some operations on fuzzy multi-hypergroups, study them, and present some examples.
Proposition 7. Let be a hypergroup and be fuzzy multisets of H. If A and B are fuzzy multi-hypergroups of H and one of them is the constant fuzzy multi-hypergroup then is a fuzzy multi-hypergroup of H.
Proof. We prove conditions of Definition 7 are satisfied for . (1) Let and . Then . Having fuzzy multi-hypergroups of H implies that and . The latter implies that . (2) Without loss of generality, let B be the constant fuzzy multiset of H with for all . Let . Then there exist such that and . The latter implies that . (3) is done in a similar way to (2). □
Example 13. Let be the hypergroup defined in Example 9 and be fuzzy multisets of H defined as:Since A is a fuzzy multi-hypergroup of H and B is a constant fuzzy multi-hypergroup of H, it follows that is a fuzzy multi-hypergroup of H. Definition 10. Let H be any set and be fuzzy multisets of H with fuzzy count functions respectively. Then the fuzzy multiset is given as follows: For all , Example 14. Let , be fuzzy multisets of H given as :Then . Proposition 8. Let be a hypergroup and be fuzzy multisets of H. If A and B are fuzzy multi-hypergroups of H and A or B is constant then is a fuzzy multi-hypergroup of H.
Proof. Without loss of generality, let B be the constant fuzzy multiset of H with for all . We prove that the conditions of Definition 7 are satisfied for . (1) Let and . Having implies that . We get that . (2) Let . Then there exist such that and . The latter implies that . (3) is done in a similar way to (2). □
Example 15. In Example 13, is a fuzzy multi-hypergroup of H.
Definition 11. Let H be a non-empty set and A be a fuzzy multiset of H. We define , the complement of A, to be the fuzzy multiset defined as: For all , Remark 4. Let be a hypergroup and A be the constant fuzzy multi-hypergroup of H defiend in Example 12. Then, is also a fuzzy multi-hypergroup of H.
We present an example of a (non-constant) fuzzy multiset A of H where A and are both fuzzy multi-hypergroups of H.
Example 16. Let , be the biset hypergroup on H and A be a fuzzy multiset of H defined as: . Proposition 2 asserts that A and are fuzzy multi-hypergroups of H.
Remark 5. Let be a hypergroup and A be a fuzzy multi-hypergroup of H. Then, is not necessary a fuzzy multi-hypergroup of H.
The following example is an illustration for Remark 5.
Example 17. Let be the hypergroup defined in Example 9 and A be the fuzzy multi-hypergroup of H defined as: . Then is not a fuzzy multi-hypergroup of H as and .
Definition 12. Let X be any set, and A a fuzzy multiset of X with fuzzy count function . Then the selection operation ⊗ is defined by the fuzzy multiset with the fuzzy count function as follows: Proposition 9. Let be a hypergroup and S be a subhypergroup of H. If A is a fuzzy multi-hypergroup of S then is a fuzzy multi-hypergroup of H.
Proof. Let . If or then . If then . Having A a fuzzy multi-hypergroup of S implies that . The latter implies that . We prove condition 2. of Definition 7 and condition 3 is done similarly. Let . If or then for all with , . If then there exist such that and . The latter implies that . □
Definition 13. Let X be any set, and A a fuzzy multiset of X with fuzzy count function . Then, the selection operation ⊙ is defined by the fuzzy multiset with the fuzzy count function . Proposition 10. Let be a hypergroup and with a subhypergroup of H. If A is a fuzzy multi-hypergroup of then is a fuzzy multi-hypergroup of H.
Proof. is given by the fuzzy count function
where
One can easily see that
. Proposition 9 completes the proof. □
Proposition 11. be hypergroups with fuzzy multisets respectively. If A and B are fuzzy multi-hypergroups of and respectively then is a fuzzy multi-hypergroup of the productional hypergrpup , where for all , .
Proof. Let
. Then
and
. Having
fuzzy multi-hypergroups of
respectively implies that
and
. We get now
The latter implies that
. We now prove condition 2. of Definition 7 and condition 3. is done similarly. Let
. Having
,
implies that there exist
such that
and
. We get that
. The latter implies that there exist
such that
and
. □
Corollary 1. be a hypergroup with fuzzy multiset for . If is a fuzzy multi-hypergroup of then is a fuzzy multi-hypergroup of the productional hypergroup , where for all , .
Proof. The proof follows by using mathematical induction and Proposition 11. □
Example 18. Let be the hypergroup defined in Example 9 and be the biset hypergroup on the set . Then, the productional hypergroup is given by the following table:Since is a fuzzy multi-hypergroup of H and is a fuzzy multi-hypergroup of J. Thenis a fuzzy multi-hypergroup of . The following propositions (Propositions 12 and 13) deal with the strong homomorphic image and pre-image of a fuzzy multi-hypergroup.
Proposition 12. Let be hypergroups, be fuzzy multisets of respectively and be a strong homomorphism. If A is a fuzzy multi-hypergroup of then is a fuzzy multi-hypergroup of ,
Proof. Let and . If or then or . We get that . If and then there exist such that and . Having f a homomorphism implies that . The latter implies that there exists such that . Since A is a fuzzy multi-hypergroup of , it follows that . We prove now condition 2. of Definition 7 and condition 3. is done similarly. Let . If or then for all such that , . If and then there exist such that and . Having A a fuzzy multi-hypergroup of implies that there exist with and . Since f is a strong homomorphism, it follows that and . □
We can use Proposition 12 to prove Proposition 9.
Corollary 2. Let be a hypergroup and S be a subhypergroup of H. If A is a fuzzy multi-hypergroup of S then is a fuzzy multi-hypergroup of H.
Proof. Let be the inclusion map defined by for all . One can easily see that is the fuzzy count function of . □
Proposition 13. Let be hypergroups, be fuzzy multisets of respectively and be a surjective strong homomorphism. If B is a fuzzy multi-hypergroup of then is a fuzzy multi-hypergroup of .
Proof. Let and . Then . Having implies that . We prove now condition 2. of Definition 7 and condition 3. is done similarly. Let . Having and B a fuzzy multi-hypergroup of implies that there exist such that and . Since f is a surjective strong homomorphism, it follows that there exist such that and . We get now that . □