1. Introduction
Symmetry is one of the central concepts of science, especially theoretical physics, mathematics and geometry of the 20th century. A given phenomenon or object is symmetrical if it is possible to introduce or consider a certain symmetry operation by which the phenomenon or object becomes in a certain sense identical to itself. The notion of symmetry has fascinated thinkers since antiquity (e.g., Pythagoreans). Later, in the so-called Erlangen program, Felix Klein tied a group of symmetry to each geometry. Mathematically, these symmetry operations are most often described by the term “group”. We distinguish continuous symmetry, which are described mathematically mainly by the term “Lie groups”, and discrete symmetry, which are described mainly by the term “discrete group”. In mathematics, a symmetric relation is one in which variables can be exchanged or index permutations can be made without changing the relation (understood as a geometric object). The natural generalization of classical group theory is the approach of algebraic hyperstructures, introduced by F. Marty [
1] during the eighth Congress of Scandinavian Mathematicians that was held in 1934. Marty generalized the notion of a group (which is a non-empty set with a binary operation satisfying some axioms and the operation of two elements is an element) to that of a hypergroup. A hypergroup is a non-empty set equipped with an associative and reproductive hyperoperation, where the composition of any two elements in it is a non-empty set. Since then, researchers started studying different kinds of hyperstructures such as: hyperrings, hypermodules, hypervector spaces, and many others by considering both parts: theoretical part as well as their applications to different subjects of science. Later in 1990, Th. Vougiouklis introduced weak hyperstructures (or
-structures) as a generalization of the concept of algebraic hyperstructures (hypergroups, hyperrings, hypermodules). The name “weak hyperstructures” is due to having some axioms of classical algebraic hyperstructures are replaced by their corresponding weak axioms in weak hyperstructures. Many researchers such as Corsini [
2], Corsini and Leoreanu [
3], Davvaz [
4,
5], Davvaz and Leoreanu-Fotea [
6], Davvaz and Cristea [
7] and Vougiouklis [
8] wrote books related to (weak) hyperstructure theory and their applications. An overview about hyperstructure theory was published by Hoskova and Chvalina in [
9].
On the other hand, fuzzy mathematics is an almost new branch in mathematics which was introduced in 1965 by Zadeh (see [
10]). It is an extension of the classical notion of set and it is related to fuzzy set theory and fuzzy logic. Fuzzy sets are sets whose elements have degrees of membership that vary between 0 and 1 both inclusive. In classical set theory, the elements’ membership in a certain set is usually identified by the condition that an element either belongs to the set or does not belong to it. By contrast, fuzzy set theory enables the gradual evaluation of the membership of elements in a set with values ranging between 0 and 1. If the membership function of a fuzzy set takes only the values 0, 1 then we go back to the classical notion of a set. As a generalization of fuzzy sets, Yager [
11] introduced the concept of Fuzzy Multiset and investigated a calculus for them. Fuzzy Multiset permits the occurrence of an element more than once and each occurrence may have the same or different membership values.
In [
12], Onasanya and Hoskova-Mayerova introduced multi-fuzzy groups induced by multisets. In [
13,
14], the authors studied fuzzy multi-polygroups and fuzzy multi-hypergroups. Moreover, Davvaz [
15] and Davvaz et al. [
16] discussed fuzzy
-ideals and generalized fuzzy
-ideals and investigated their properties. Our paper generalizes the work in [
12,
13,
15,
17] to combine
-rings and fuzzy multisets. More specifically, it is concerned about fuzzy multi-
-ideals and generalized fuzzy multi-
-ideals and it is constructed subsequently: Our motivation is described in Introduction,
Section 2 presents basic notions with respect to (weak) hyperstructures and fuzzy multisets that are used throughout the paper.
Section 3 defines and studies the properties of fuzzy multi-
-ideals and their relation to
-ideals. Finally,
Section 4 defines generalized fuzzy multi-
-ideals and studies their properties.
3. Fuzzy Multi--Ideal
In this section, we introduce for the first time the notion of fuzzy multi-
-ideal as a generalization of fuzzy
-ideal, present several examples and results related to this new concept. The results in [
15] related to fuzzy
-ideals can be considered as a special case of the results of this section.
Definition 4. Let be an -ring. A fuzzy multiset A (with fuzzy count function ) over R is a fuzzy multi--ideal of R if for all , the following conditions hold.
- 1.
;
- 2.
for every there exists such that and ;
- 3.
for every there exists such that and ;
- 4.
for all .
Remark 1. Let be an -ring with “+” a commutative hyperoperation and A be a fuzzy multiset over R. To prove that A is a fuzzy multi--ideal of R, it suffices to prove Conditions 1, 2, and 4 or Conditions 1, 3, and 4 of Definition 4. This is clear as in the case of commutative -group, Conditions 2 and 3 are equivalent to each other.
Example 3. Let be an -ring with a fixed element and A be a fuzzy multiset of R defined as for all . Then A is a fuzzy multi--ideal of R (the constant fuzzy multi--ideal.).
Remark 2. Let be an -ring. Then we can define at least one fuzzy multi--ideal of R which is mainly the one that is defined in Example 3.
We present some examples on non-constant fuzzy multi--ideals.
Example 4. Let be the -ring defined as follows:It is clear that is a fuzzy multi--ideal of . Example 5. Let be the -ring defined by the following tables:It is clear that is a fuzzy multi--ideal of . Example 6. Let be the -ring defined by the following tables:It is clear that both: and are fuzzy multi--ideals of .
Proposition 1. Let be an -group and “·” be any hyperoperation on R with for all . Then A is a fuzzy multi–-ideal of the -ring if and only if A is the constant fuzzy multi--ideal of R.
Proof. It is clear that if A is the fuzzy multiset described in Example 3 then A is a fuzzy multi--ideal of R. Let A be a fuzzy multi--ideal of R and . Having for all and Condition 4 of Definition 4 implies that both and are greater than or equal . Thus, for all . ☐
Example 7. Let be the -ring defined by the following tables:Using Proposition 1, we get that the constant fuzzy multi--ideal of R is the only fuzzy multi--ideal of R. Notation 1. Let be an -ring, A be a fuzzy multiset of R and . Then
if ,
if ,
if where
Definition 5. Let be an -ring and A be a fuzzy multiset of R. Then and .
Proposition 2. Let be an -ring and A be a fuzzy multi--ideal of R. Then is either the empty set or an -ideal of R.
Proof. Let . First, we show that . We prove and is done similarly. For all and , we have . The latter implies that and hence, . Moreover, for all , Condition 2 of Definition 4 implies that there exist such that and . The latter implies that and . Thus, . Now, we prove that and . We prove that and is done similarly. Let and . Then for all , Condition 4 of Definition 4 implies that . Thus, . ☐
Proposition 3. Let be an -ring and A be a fuzzy multi--ideal of R. Then is either the empty set or an -ideal of R.
Proof. Let . First, we show that . We prove and is done similarly. For all and , we have . The latter implies that and hence, . Moreover, for all , Condition 2 of Definition 4 implies that there exist such that and . The latter implies that and . Thus, . Now, we prove that and . We prove that and is done similarly. Let and . Then for all , Condition 4 of Definition 4 implies that . Thus, . ☐
Example 8. Let be the -ring presented in Example 6. Having
, fuzzy multi--ideals of , we get that and are -ideals of . Also, .
Notation 2. Let be an -ring, A be a fuzzy multiset of R and . We say that if and for all . If and then we say that and are not comparable.
Theorem 1. Let be an -ring, A a fuzzy multiset of R with fuzzy count function and where for and . Then A is a fuzzy multi--ideal of R if and only if is either the empty set or an -ideal of R.
Proof. Let be an -ideal of R and . By setting , we get that . Having an -ideal of R implies that for all , . We prove Condition 2 of Definition 4 and Condition 3 is done similarly. Let and . Then . Having an -ideal of R implies that . The latter implies that there exist such that . Thus, . We prove now Condition 4 of Definition 4. Let and . By setting and , we get that and . Having and implies that and . Thus, .
Conversely, let A be a fuzzy multi--ideal of R 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--ideal of R, it follows that there exist such that and . The latter implies that and hence, . We prove now that and is done similarly. Let and . For all , Condition 4 of Definition 4 implies that . Thus, . ☐
Corollary 1. Let be an -ring. If R has no proper -ideals then every fuzzy multi--ideal of R is the constant fuzzy multi--ideal.
Proof. Let A be a fuzzy multi--ideal of R and suppose, to get contradiction, that A is not the constant fuzzy multi--ideal. Then there exist with . We have three cases for : , , and and are not comparable. If then and for . If or and are not comparable, then and for . Using Theorem 1, we get that is an -ideal of R. ☐
Proposition 4. Let be an -ring and S be an -ideal of R. Then for some where for and .
Proof. Let
where
for
and define the fuzzy multiset
A of
R as follows:
It is clear that
. We still need to prove that
is a fuzzy multi-
-ideal of
R. Using Theorem 1, it suffices to show that
is an
-ideal of
R for all
with
and
for
. One can easily see that
Thus,
is either the empty set or an
-ideal of
R. ☐
Next, we deal with some operations on fuzzy multi--ideals.
Definition 6. Let be an -ring and be fuzzy multisets of R. Then is defined by the following fuzzy count function. Theorem 2. Let be an -ring and A be a fuzzy multiset of H. If A is a fuzzy multi--ideal of R then .
Proof. Let . Then for all . The latter implies that . Thus, . Having an -ring and A a fuzzy multi--idear of R implies that for every there exist such that and . Moreover, we have . Thus, . ☐
Definition 7. Let R be a non-empty set and A be a fuzzy multiset of R. We define , the complement of A, to be the fuzzy multiset defined as: For all , Example 9. Let be a set and A be a fuzzy multiset with fuzzy count function defined as: . Then .
Remark 3. Let be an -ring and A be the constant fuzzy multi--ideal of R defined in Example 3. Then is also a fuzzy multi--ideal of R.
Remark 4. Let be an -ring and A be a fuzzy multi--ideal of R. Then is not necessary a fuzzy multi--ideal of R.
We illustrate Remark 4 by the following example.
Example 10. Let the triple be the -ring defined in Example 6 and be a fuzzy multi--ideals of .
Then is not a fuzzy multi--ideals of . This is clear as and .
Proposition 5. Let be an -ring with a fuzzy multiset for all . If is a fuzzy multi--ideal of for all then is a fuzzy multi--ideal of the . Where .
Proof. The proof is straightforward. ☐
We present an example when .
Example 11. Let be the -ring presented in Example 4 andbe a fuzzy multi--ideal of . Then given by:is a fuzzy multi--ideal of . The next two propositions discuss the strong homomorphic image and pre-image of a fuzzy multi--ideal.
Proposition 6. Let be -rings, A be a fuzzy multiset of and be a surjective strong homomorphism. If A is a fuzzy multi--ideal of then is a fuzzy multi--ideal of .
Proof. Let and . Since and , it follows that 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--ideal of , it follows that . We prove now Condition 2 of Definition 4 and Condition 3 is done similarly. Let . Since and then there exist such that and . Having A a fuzzy multi--ideal of implies that there exist with and . Since f is a strong homomorphism, it follows that and . We prove now Condition 4 of Definition 4 for . Let and . Since and , it follows that there exist such that and . Having f a strong homomorphism implies that . The latter implies that there exists such that . Since A is a fuzzy multi--ideal of , it follows that . ☐
Proposition 7. Let be -rings, B be a fuzzy multiset of and be a surjective strong homomorphism. If B is a fuzzy multi--ideal of then is a fuzzy multi--ideal of .
Proof. Let and . Then . Having implies that . We prove now Condition 2 of Definition 4 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 . To prove Condition 4 for , let . Then . Having completes the proof. ☐
Corollary 2. Let be an -ring with fundamental relation and A be a fuzzy multiset of R. If A is a fuzzy multi--ideal of R then B is a fuzzy multi--ideal of . Where Proof. Let
A be a fuzzy multi-
-ideal of
R and
be the map defined by
. Then
f is a surjective homomorphism. Proposition 6 asserts that
is a fuzzy multi-
-ideal of
where
Therefore, B is a fuzzy multi--ideal of . ☐
Definition 8. Let be a ring. A fuzzy multiset A (with fuzzy count function ) over R is a fuzzy multi-ideal of R if for all , the following conditions hold.
- 1.
for all ;
- 2.
for all ;
- 3.
for all .
Proposition 8. Let be an -ring with fundamental relation and A be a fuzzy multiset of R. If A is a fuzzy multi--ideal of R then B is a fuzzy multi-ideal of the ring . Where Proof. Corollary 2 asserts that Conditions 1 and 3 of Definition 8 are satisfied. We need to prove Condition 2. Having a ring implies that there exist a zero element, say such that and for all . Having B a fuzzy multi--ideal of implies that for all . Since a ring, it follows that for every there exist with . Having B a fuzzy multi--ideal of and using Condition 2 of Definition 4 implies that for and there exists such that and . It is clear that . ☐
Example 12. Let be the -ring presented in Example 6. One can easily see that the fundamental ring and is isomorphic to the ring of integers under standard addition and multiplication modulo 2. Using Proposition 8, we get that is a fuzzy multi-ideal of .
4. Generalized Fuzzy Multi--Ideal
In this section, we generalize the notion of fuzzy multi-
-ideal defined in
Section 3 to generalized fuzzy multi-
-ideal, investigate its properties, and present some examples.
Notation 3. Let A be a fuzzy multiset of a non-empty set R with a fuzzy count function . We say that:
- 1.
when ,
- 2.
when ,
- 3.
when or ,
- 4.
where
Definition 9. Let be an -ring. A fuzzy multiset A (with fuzzy count function ) over R is an -fuzzy multi--ideal of R if for all , , the following conditions hold.
, implies for all ;
, implies for some with ;
, implies for some with ;
, implies for all
(, implies for all ).
Remark 5. Let be an -ring and A a fuzzy multiset of R. If A is a fuzzy multi--ideal of R then A is an -fuzzy multi--ideal of R.
Example 13. Let be any -ring. Then the constant fuzzy multiset of R is an -fuzzy multi--ideal of R.
Example 14. Let be the -ring presented in Example 4. Having is a fuzzy multi--ideal of implies that is an -fuzzy multi--ideal of .
The converse of Remark 5 does not always hold. We illustrate this idea by the following example.
Example 15. Let be the -ring defined by the following tables: One can easily see thatis an -fuzzy multi--ideal of R but not a fuzzy multi--ideal of R. This is clear as but . Proposition 9. Let with and . If then there exists such that .
Proof. We have the following cases:
Case . Take .
Case . Then there exists with . Since are real numbers, it follows that there exists a real number with . By taking , we get that . ☐
Theorem 3. Let be an -ring, A a fuzzy multiset of R with fuzzy count function , and for all , and are comparable. If A is an -fuzzy multi--ideal of R then the following conditions hold:
- (a)
for all ;
- (b)
For all there exists such that and - (c)
For all there exists such that and - (d)
For all , and .
Proof. It suffices to show that , , , and .
(1) → (a): Let . Since each of are comparable with , we can consider the cases: and .
For the case , suppose that there exists with . We get that . Proposition 9 asserts that there exists r with . The latter implies that and . Moreover, having implies that . We get that which contradicts (1).
For the case , suppose that there exists with . We get that and . It is clear that which contradicts (1).
(2) → (b): Let . Since each of are comparable with , we can consider the cases: and .
For the case , suppose that for all with we have . Proposition 9 asserts that there exists r with . It is clear that and . The latter contradicts (2).
(3) → (c): This case is done in a similar manner to that of (2) → (b).
(4) → (d): Let . Since is comparable with , we can consider the cases: and .
For the case , suppose that there exists with . Proposition 9 asserts that there exists r with . Then and which contradicts (4).
For the case , suppose that there exists with . Then and which contradicts (4). ☐
Remark 6. Theorem 3 can be used only when and are comparable. Otherwise, we should use Definition 9.
Note that according to Remark 6, we can not use Theorem 3 to the fuzzy multiset given in Example 5 as is not comparable with .
Remark 7. In case of fuzzy -ideal of an -ring, the conditions of Theorem 3 are necessary and sufficient for a fuzzy set to be a fuzzy -ideal (see [16]). Whereas in our case (fuzzy multiset), the converse of Theorem 3 is not always true. (See Example 16). Example 16. Let be the -ring defined in Example 15 and let A be the fuzzy multiset of R with count function defined by: , , . Having , it is easy to see that Conditions (a), (b), (c), and (d) of Theorem 3 are satisfied. But A is not an -fuzzy multi--ideal of R. By taking , we get that . Having , and implies that Condition 4 of Definition 9 is not satisfied.