Abstract
The hesitant fuzzy set model has attracted the interest of scholars in various fields. The striking framework of hesitant fuzzy sets is keen to provide a larger domain of preference for fuzzy information modeling of deployment membership. Starting from the hybrid properties of hesitant fuzzy ideals (HFI), this paper constructs a new generalized hybrid structure -HFI. The concept of -hesitant fuzzy exchange ideal in -algebra is considered. Lastly, -hesitant fuzzy exchange ideal features are described.
1. Introduction
When dealing with information on all aspects of uncertainty, nonclassical logic always makes use of classical logic. Nonclassical logic is a useful tool in computer science because it deals with fuzzy information and uncertainty. In the literature, the study of -algebras was first proposed by Imai and Iséki [1] in 1966 and such algebras can be regarded as a generalization of propositional logic. The study -algebras have been developed by many people and have been extended to the fuzzy setting. After the introduction of fuzzy sets introduced by Zadeh [2], there have been many generalizations of this fundamental concept. In 2010, Torra [3] considered hesitant fuzzy sets. The hesitant fuzzy set model is useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision-makers.
Algebraic structures provide sufficient motivation for researchers to examine various concepts and stem from the broader field of abstract algebra blur set frame. In 2011, Xia and Xu [4] described hesitant fuzzy information aggregation techniques, and this concept was applied to -algebras, -algebras, residuated lattices, -algebras, and -algebras [5–9]. Jun and Ahn [6] investigated the concept of hesitant fuzzy subalgebras and HFIs of -algebras. In 2018, Alshehri et al. [10] put forward the concept of new types of HFIs in -algebras. As a continuation of this study, we describe certain concepts, including -HFIs and-hesitant fuzzy commutative ideals in -algebras.
2. Basic Notions
A set with a constant element 0 and a binary operation is said to be a -algebra [1] if it satisfies the axioms:
For all ,
In a -algebra , we can define the relation by if and only if .
Then, is a partially ordered set with the least element . In any -algebra , the following properties hold:
implies and for all .
Let be a -algebra and let be a nonempty subset of . Then, is called an ideal of [11] if it satisfies the following: (1) (2) and imply that for all
A subset of a -algebra is called a commutative ideal [12] of if it satisfies the following: (1) (2) and imply that for all
A fuzzy set in is said to be a fuzzy ideal of if it satisfies the following: (1) for all (2) for all
Let be a reference set and be a nonempty subset of , a hesitant fuzzy set.
on [3] satisfying the following condition: is called a hesitant fuzzy set related to (briefly, -hesitant fuzzy set) on and is represented by , where is a mapping from to with for all .
Let be a reference set and be a nonempty subset of , an -hesitant fuzzy set of is called a HFI [6] of related to (briefly, -HFI of if it satisfies the following: (1) for all (2) for all
Given a nonempty subset of , an -hesitant fuzzy set of is called a hesitant fuzzy commutative ideal [10] of related to (briefly, -hesitant fuzzy commutative ideal of ) if it satisfies
for all
for all
Let be a nonempty finite universe and be a nonempty set. A -hesitant fuzzy set is a set given by where . The function is called the membership function of -hesitant fuzzy set, and the set of all -hesitant fuzzy set over will be denoted by .
Let be a hesitant fuzzy set of a -algebra . The set where is called a hesitant fuzzy -level set of .
Theorem 1 (see [6]). For a subalgebra, of a -algebra , every -HFI is an -hesitant fuzzy subalgebra.
Proposition 2 (see [13]). In -algebra the following conditions hold, for all ,
3. -Hesitant Fuzzy Ideals
Definition 3. Let be a nonempty finite universe, be a nonempty set and be the subset of , a -hesitant fuzzy ideal (briefly: -ideal) if it satisfies the following assertion: (1)(2)
Example 1 Denote . The binary operation on is given by Cayley (Table 1).
For a subset . Let be a of defined by
Then, is a -ideal of .
Proposition 4. Let be a subset of and be a -HFI of . Then, the following assertions are valid: (1) for all (2) for all
Proof. (1)Suppose implies (for all ) and so by (2)Suppose implies (for all ) so It follows that
Proposition 5. Every A -ideal of satisfies the following condition: (1) with for all (2) with for all
Theorem 6. If a -HFI of , then for any , and
Theorem 7. Let be a -HFI of . Then, the following are equivalent: (i) for all (ii) for all
Proof. Suppose condition is valid. Since
Applying, by Proposition 2 and , we have
Hence, condition holds
Suppose condition is valid. If we put in then
hence, the condition holds.
The proof is complete.
Theorem 8. Let be a -HFI of , then the set is an ideal of for all .
Proof. Let be such that and . Then, It follows from that
So that and , therefore, is an ideal of for all .
Theorem 9. Suppose that is a -hesitant fuzzy set of , where is a nonempty subset of . Then, the following are equivalent: (i) is a -HFI of (ii)For any , the set is an ideal of
Proof. Assume that is a -ideal of . Let and be such that and Then,
It follows that
Hence, and
Therefore is an ideal of
Suppose that is an ideal of . For any , let
Then . Since is an ideal of . we have
Let and . Then and such that , which imply that . Thus,
Therefore, is a -ideal of .
4. -Hesitant Fuzzy Commutative Ideals
Definition 10. Let be a universal set and be a nonempty set. A -hesitant fuzzy commutative ideal (-HFCI) of if it satisfies the following assertion: (1) (2)
Example 1. Let be a set with the binary operation which is defined in Cayley (Table 2).
Let such that . We define a of as follows:
where . By direct calculations, one can see that is -ideal of
Theorem 11. Every -hesitant fuzzy CI is a -hesitant fuzzy ideal of
Proof. Assume that is a -ideal of a -algebra , for any . We have Therefore, is a -ideal in .
The following example shows that the converse of theorem 6 is not true.
Example 2. Let be a set with the Cayley (Table 3).
Let such that We define a mapping
where . It is routine to verify that is -ideal of . But it is not a -ideal of . Since
Theorem 12. Let be a -HFI of a -algebra . Then, is a -hesitant fuzzy CI of if and only if it satisfies the following condition:
Proof. Assume that is -ideal. Taking in and using Also, we use .
Conversely, Let As be a -ideal of satisfying condition (1).
Then,
combining (1) and (2), then we obtain .
The proof is complete.
Lemma 13. Any -HFI of a -algebra satisfies
Proof. Assume that holds. Then, It follows that The proof is complete.
Theorem 14. For any commutative in a -algebra . Every -HFI is commutative.
Proof. Let be a -ideal of a commutative -algebra . It is sufficient to show that satisfies condition .Let . Then, That is, By Lemma 13, we have Thus, holds. Therefore, is a -HFCI.
Definition 15. Let be a -hesitant CI of a -algebra , for , the set of a CI is called -hesitant -level CI of .
Theorem 16. In -algebra , any CI of can be realized as -hesitant -level CI of some -HFCI of .
Proof. Let be a CI of -algebra and let be a -hesitant fuzzy set of defined by
where . Let .
If and then Thus,
and so
(i)If and then Hence,
(ii)If exactly one of and belongs to , then exactly one of and is equal to zero. So,The results above show
It is clear that for all . Therefore, is a -ideal of . Obviously, .
Theorem 17. If a -ideal of a -algebra . Then, two-level CI and where of are equal if and only if there is no such that .
Proof. Let If there exists such that , then . This is impossible. Conversely, assume that there is no such that .
implies If then and so because .
Hence, which says that .
Thus, .
This completes the proof.
Let be a -hesitant fuzzy set in and let denote the image of .
Theorem 18. Let be a -algebra and a -HFCI of . If where , then the family of CIs constitutes all the level CIs of
Proof. Let and If then . Since , we have and
If , then there is no such that . From above theorem 10, it follows that . This shows that for any with , the level CI is in .
Lemma 19. Given a -algebra and a -ideal over . If and belong to such that , then .
Proof. Assume that , say . Then, there is such that , and so and . Thus, , which is a contradiction to our fact. This completes the proof.
5. -Hesitant Fuzzy Characteristic CIs
A mapping of a -algebra is called a homomorphism if satisfying the identity for all . Throughout, will denote the -algebra of automorphisms of .
Definition 20. Let be a homomorphism of -algebras. For any -ideal of , we define a new -ideal in by
Theorem 21. Let be a homomorphism of -algebra . If is a -ideal of , then is a -ideal of .
Proof. Let Let
Definition 22. A CI of a -algebra is called a characteristic CI (CCI) of if for all .
Definition 23. A -HFCI of a -algebra is called a -hesitant fuzzy CCI of if
Theorem 24. Let be a -hesitant fuzzy characteristic CI of . Then, each -level CI of is a characteristic commutative ideal of .
Proof. Assume and . Since is a -hesitant fuzzy characteristic commutative ideal of , we have .
It follows that and hence .
To show the reverse inclusion, let and let be such that . Then, whence . It follows that so that . Thus, is a CCI of .
The proof of the following lemma is obvious, and we omit the proof.
Lemma 25. Let be a -HFCI of and let . Then, if and only if and , for all .
Now, we consider the inverse of Theorem 24
Theorem 26. Let be a -HFCI of . If each level CI of is a CCI of , then is a -hesitant fuzzy characteristic commutative ideal of .
Proof. Let and . Then, and for all , by Lemma 25 Since by hypothesis, we have and hence . Let . If possible, let .
Then, . Since { is one to one, it follows that , which is a contradiction. Hence, . It follows that is a -hesitant fuzzy CCI of . This completes the proof.
6. Conclusions
A new concept of HFI is considered by applying a two-dimensional membership function, namely, -HFI. Several properties and theorems of -HFI are proved. In this regard, we propose the concept of -HFCI in -algebra and prove some related properties. We have considered the features of -HFCI. We study some feature properties related to -HFCI. Our future research is to find ways to apply -HFI to a wide range of logical algebraic systems, such as pseudo--algebras [14, 15]. For other notions, the readers are suggested to see [16–28].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University, for the funding publication of this project.