Abstract
This paper deals with the study of generalizations of fuzzy subalgebras and fuzzy ideals in BCK/BCI-algebras. In fact, the notions of -fuzzy subalgebras, -fuzzy ideals, and -fuzzy ideals in BCK/BCI-algebras are introduced. Some examples are provided to demonstrate the logic of the concepts used in this paper. Moreover, some characterizations of these notions are discussed. In addition, the concept of -fuzzy commutative ideals is introduced, and several significant characteristics are discussed. It is shown that for a BCK-algebra , every -commutative ideal of a BCK-algebra is an -fuzzy ideal, but the converse does not hold in general; a counter example is constructed.
1. Introduction
To deal with possible complexity associated with expectations, state imprecision, and desires, a fuzzy set is a valuable method, as proposed by Zadeh [1]. Fuzzy set theory has since become an active research area in a different field. The idea of quasicoincidence of a fuzzy point with a fuzzy set, as stated in [2], was crucial in generating some different types of fuzzy subgroups, known as -fuzzy subgroups, as introduced by Bhakat and Das in [3]. The -fuzzy subgroup, in particular, is an important and valuable generalization of Rosenfeld’s fuzzy subgroup. The idea of -fuzzy subalgebras in BCK/BCI-algebras is also interesting and useful generalizations of some well-known ideas of fuzzy subalgebras (see, for example, [4–11]). Several similar ideals based on fuzzy subalgebras and ideals have recently been studied in various algebras (see, for example, [12–18]). Many researchers have also extended fuzzy set theory and related concepts to different fields (see, e.g., [19–24]).
Motivated by a lot of work on ideal theory in BCK/BCI-algebras based on the fuzzy set theory, it is reasonable to introduce a generalized version of the fuzzy ideals of BCK/BCI-algebras. To that aim, in Section 2, some basic definitions and elementary results are presented. Then, in Section 3, we present the concepts -fuzzy subalgebras, -fuzzy ideals, and -fuzzy ideals and associated characteristics are investigated. In Section 4, the concept of -fuzzy commutative ideals is presented and various properties are investigated, as well as their relationship with -fuzzy ideals.
2. Preliminaries
An algebra is called a BCI-algebra if , (1)(2)(3)(4) and
If satisfies (2), (3), (40), (43), and , then is a BCK-algebra.
Any BCK-algebra satisfies the following: (5)(6)
Define a partially ordered on as .
A subset of is said to be a subalgebra if , and is said to be an ideal of if and implies .
A mapping is said to be a fuzzy set (briefly, FS) of . If , , then is said to be a fuzzy subalgebra. If and , , then is said to be a fuzzy ideal.
Definition 1. Let and . An ordered fuzzy point (briefly, OFP) of is defined as .
Clearly, is a FS of . For FS of , we represent as in the sequel. So, .
Lemma 2 (see [4]). A FS in is an -fuzzy subalgebra of it satisfies
Lemma 3 (see [25]). A FS in is an -fuzzy subalgebra of it satisfies
3. -Fuzzy Ideals
In what follows, denotes a BCK/BCI-algebra unless otherwise specified.
Definition 4. Let be an OFP of and . Then, is said to be -quasicoincident with a FS of , represented as , if .
Let us take, . For an OFP , we define
(1) if (2) if or (3) if does not hold for ,
Definition 5. A FS of is said to be an -fuzzy subalgebra (in short -FSA) of if and imply , , and .
Example 6. Consider a -algebra with operation , which is described in Table 1. Define by the following: Take “” and “.” Then, it yields that is an -FSA of .
Theorem 7. A FS of is an -FSA of .
Proof. On the contrary, assume that for some . Choose such that
Then, and , but , which is not possible. Hence, .
() Assume that , . Let and , . Then, and . So, . If , then imply that . If , then . So, implies that . Hence, . Therefore, is an -FSA of .
Definition 8. A FS of is said to be an -fuzzy ideal (in short -FI) of if (1) implies , and(2) and imply and .
Example 9. Take a -algebra with operation which is described in Table 2. Define by the following: Consider “” and “.” It is easy to check that is an -FI of .
Definition 10 (see [5]). A FS of is said to be an -fuzzy ideal (briefly -FI) of if: (1) implies , and(2) and imply and .
Proposition 11. In , every -FI is -FI.
Proof. Suppose that is any -FI of . Take for and . Then, by hypothesis, . It implies that or , and so or . Therefore, . Next, let and . So, implies or . Therefore, or . Thus, . Hence, is an -FI of .
Remark 12. In general, the converse of Proposition 11 is not valid. The following example demonstrates this.
Example 13. Take a -algebra with operation which is described in Table 3.
Define by
Take and . Then, is an -FI of but it is not an -FI of as and but .
Definition 14. A FS of is said to be an -fuzzy ideal (briefly -FI) of if: (1) implies , and(2) and imply and .
Lemma 15. In , every -FI is -FI.
Proof. Suppose that is an -FI of . Let for and . Then, . So, by hypothesis, . Suppose that and . Then, and . Therefore, by hypothesis, . Hence, is an -FI of .
Remark 16. In general, the converse of Lemma 15 is not valid. This is illustrated by the following.
Example 17. From Example 13, define by Take “” and “.” Then, is an -FI of but it is not an -FI of as and but .
Lemma 18. Let be a FS of . Then, .
Proof. Contrary assume that, for some , . Take such that
Then, , but , a contradiction. Hence, .
() Let such that . Then, . So,
Now, if , then . Therefore, . Again, if , then . So, . This follows that . Hence, .
Lemma 19. Let be FS of . Then, and imply .
Proof. On contrary suppose that for some . Choose such that . Then, , , but , which is not possible. Thus, we have shown that () Let and , . Then, and . Thus, Now, if , then , then ; otherwise, i.e., when , then So, we have This implies that . Hence, , as required.
We obtain the following theorem by combining Lemmas 18 and 19.
Theorem 20. A FS of is an -FI of if and only if (1)(2)
Lemma 21. Let be an -FI of such that . Then, .
Proof. Let for . Then, . By hypothesis, we have
Lemma 22. Let be an -FI of . Then, for any ,
Proof. Let for . We have
Theorem 23. Every -FI of -algebra is an -FS of .
Proof. Let be an -FI of and . As in , so by Lemma 21, we have Since is an -FI of , we have Hence, is an -FS of .
Remark 24. In general, the converse of Theorem 23 is not valid.
Example 25. Take a -algebra with operation which is described in Table 4.
Consider the -FS of , where is defined by
Take “” and “.” Then, is not an “”-FI of as and but .
Theorem 26. Let be an -FS of . Then, is an -FI such that implies .
Proof. Follows from Lemma 22.
() Let be an -FS such that with implies . As , so by hypothesis
Hence, is an -FI of .
Theorem 27. A FS is an -FI of the set is an ideal of , .
Proof. Let such that . By Theorem 20, we have
with . It follows that . Therefore, .
Next, suppose that and . Then, and . Again, by Theorem 20, we have
Therefore, . Hence, is an ideal of .
() Suppose that is an ideal of , . If for some . Then, such that . It follows that but , a contradiction. Therefore, . Also, if for some . Then, such that
It implies that and but , a contradiction. Therefore, . Hence, is an -FI of .
Definition 28. Let be an FS of . The set is said to be an -level subset of .
Theorem 29. Let be an FS of . Then is an -FI of the -level subset of is an ideal of , .
Proof. Suppose that is an -FI of . Take any . Then, . So, or . Now, by Theorem 20, we have . Thus, when . If , then implies . Also, if , then implies . Similarly, when .
Next, take any and . Then, and , i.e, either or and either or . By assumption, . Thus, we have
Case 1. Let and . If , then
and so, . If , then
So . Hence, .
Case 2. Let and . If , then
i.e., and, thus, . If , then
and so . Hence, .
Similarly, for other cases, i.e., when , and , , we have . Therefore, for each case, , and hence .
() Let be an ideal of , On contrary, let
with . Then, such that . It implies that but , which is impossible. Therefore,
Also, if for some . Then, such that
It follows that and , but , a contradiction. Therefore, . Hence, is an -FI of .
4. -Fuzzy Commutative Ideals
Throughout this section, will stand for a BCK-algebra.
Definition 30. Let be a BCK-algebra. An FS is said to be an -fuzzy commutative ideal (briefly, -FCI) if (1) implies , and(2) and imply and .
Example 31. Take a -algebra with operation which is described in Table 5.
Define by
Take “” and “.” Then, it is easy to get, is an -FCI of .
Theorem 32. A FS of a BCK-algebra is an -FCI of (1), and(2)
Proof. Lemma 18 gives us condition (2). To prove that (3) holds in , assume that (3) does not hold in , so we have
for some . Choose such that . Then, and , but , which is not possible. Thus,
() Assume that conditions (2) and (3) hold in . From condition (2) and Lemma 18, implies . Let and , . Then, and . Thus,
Now, if , then implies ; otherwise, i.e., when , then So, we have
This implies that . Therefore, . Hence, is an -FCI of .
Theorem 33. Every -FCI of BCK-algebra is an -FI of .
Proof. Let be an -FCI of and . So, we have Hence, is an -FI of .
Remark 34. The converse of Theorem 33 is not valid in general.
Example 35. Take a -algebra with operation which is described in Table 6. Take and . Then, is an -FI of but it is not an -FCI of as and but .
Theorem 36. Let be an -FI of BCK-algebra . Then, is an -FCI of ,
Proof. Let be an -FCI of . Then, , we have By taking , we have () Assume that equation (40) holds in . Let . As is an -FI of , so we have By assumption and (43), we have Hence, is an -FCI of .
5. Conclusion
The main purpose of the present paper is to introduce the concepts of -fuzzy subalgebras and -fuzzy ideals in BCK/BCI-algebras. We provided some equivalent conditions and different characterizations of the -fuzzy ideals in terms of level subsets and -level subsets of BCK/BCI-algebras. It has been shown that in any BCK/BCI-algebras the -fuzzy ideals are -fuzzy subalgebras but the converse does not hold and an example provided in this aim. Furthermore, -fuzzy commutative ideals in BCK-algebras is introduced and some related properties of -fuzzy ideals and -fuzzy commutative ideals are considered. We hope that this work will provide a deep impact on the upcoming research in this field and other fuzzy algebraic studies to open up new horizons of interest and innovations. In future study, these notions may be extended to different algebras such as rings, hemirings, -semigroups, semihypergroups, semihyperrings, BCK/BCI-algebras, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, and EQ-algebras. Some important issues for future work are (1) to develop strategies for obtaining more valuable results and (2) to apply these notions and results for studying related notions in other algebraic (fuzzy) structures.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the manuscript.
Acknowledgments
This research has been funded by the Research Deanship at the University of Ha’il-Saudi Arabia through project number RG-20 189.