Abstract
In this paper, we propose the concept of -fuzzy p-ideals in “-algebras.” We show that “-fuzzy p-ideals” and “-fuzzy -ideals” are “-fuzzy -ideals.” However, the converse is not true, then presented examples. For a BCI-algebra , it has been shown that every -fuzzy p-ideal of is an -fuzzy ideals of but not conversely, and then, an example is given. Furthermore in , a connection between -fuzzy p-ideals and p-ideals is established.
1. Introduction
The concepts of BCK and BCI-algebras were first introduced by Imai and Ise’ki in 1966 [1, 2]. The algebraic formulations of the BCK and BCI systems are BCK and BCI-algebras in combinatory logic. Eventually, the theory of these algebras has been developed rapidly and successfully with a specific focus on the ideal theory, for instance, Liu et al. [3] studied -ideals while fuzzy -ideals are given in [4], and hybrid ideals are considered by Muhiuddin et al. [5, 6] in BCK/BCI-algebras. Recent research focused on several kinds of related ideals are studied in [7–10].
The theory of fuzzy set is given in [11] as a new discipline. Jun [12] initiated the study of fuzzy p-ideals in BCI-algebras and studied their various characteristics. Touqeer and Cagman [13] have given the notion of intuitionistic fuzzy p-ideals of BCI-algebras. Muhiuddin [14] investigated p-ideals of BCI-algebras related with neutrosophic N-structures.
In order to develop various kinds of fuzzy subgroups, the idea of “quasi-coincidence” of a fuzzy point with a fuzzy set is established in [15]. The same concepts were introduced and investigated by Jun [16, 17] in BCK/BCI-algebras. Zhan et al. [18] gave the idea of -fuzzy ideal of BCI-algebra and explored their interesting results. Also, Zhang et al. [18] applied the idea of a quasicoincidence of a fuzzy point with a fuzzy set and introduced the concepts of -fuzzy p(q and a)-ideals in BCI-algebras, while Ma et al. [19] present the ideas of various kinds of fuzzy ideals based on -interval-valued fuzzy structures.
Al-Masarwah and Ahmad [20] developed the ideas of -polar -fuzzy ideals. Takallo et al. defined and presented m-polar -fuzzy p-ideals in [21]. Numerous algebraic systems have been exposed to these structures, with a variety of outcomes [22–25].
The concept of generalized notion is natural to introduce. To do so, we introduced the concept of -fuzzy p-ideals. Furthermore, we presented the relationship between -fuzzy p-ideals and -fuzzy -ideals. Besides, we investigated the correspondence among these notions.
2. Preliminaries
An algebra ” is a BCI-algebra if , (1),(2),(3),(4) and .
A partially ordered “ on is defined as .
From now we mean as a BCI-algebra. By a fuzzy subset (in brief, FS), we mean a function .
Definition 1 (see [26]). Let and . The “ordered fuzzy point” (in brief, OFP) of is given as: It is obvious that is an FS of . In the sequel, we indicate as for any FS . In other words, .
Definition 2 (see [26]). A FS of is called an -fuzzy subalgebra (in brief, -FSA) of if and implies and .
Definition 3 (see [26]). A FS of is said to be an -FI (briefly, fuzzy ideal) of if (1) imply , and(2) and imply , and .
Lemma 4 (see [26]). Let be a FS of . Then, implies .
Lemma 5 (see [26]). Let be an -FI in such that . Then,
Lemma 6 (see [26]. Let be an -FI of . Then, , .
3. -Fuzzy p-Ideals
Definition 7. Let be in OFP of and . Then, is called -quasicoincident with a FS of , denoted as , if Suppose that . For OFP , we define (1), if (2), if or (3), if does not hold for .
Definition 8. A FS of is called an -fuzzy p-ideal (in brief, -FPI) of if (1) imply (2) and imply and .
Example 9. Consider a BCI-algebra , defined by Table 1.
Define a FS on as
It is easy to evaluate that is an -FPI for and of .
Definition 10. A FS of is called an -FPI (briefly, fuzzy p-ideal) of if (1) imply (2) and imply and .
Theorem 11. In , every -FPI is an -FPI, but converse may not be true in general.
Proof. Assume that is an -FPI of . Take for and . So by hypothesis, . It implies that or , and so, or . Thus, . Further, take any and . So, implies or . Therefore, or . Thus, , as required.
Example 12. Consider a BCI-algebra which is defined by Table 2.
We define a FS
It is easy to evaluate that is an -FPI of but not an -FPI as and but , where and .
Definition 13. A FS of is said to be an -FPI of if (1) imply (2) and imply and .
Lemma 14. In , every -FPI is -FPI.
Proof. Let be any -FPI of . Take any for and . Then, . Therefore, by hypothesis, . Assume that and for any . Then, and . So, , as required.
Example 15. Consider a BCI-algebra which is defined by Table 3:
Define by
is an -FPI of with and , although it is not an “-FPI” of as and but .
Lemma 16. Let be a FS of . Then, , , and imply .
Proof. () Contrary assume that for some , . Take s.t. . Then, and , but , which is impossible. Hence,
() Let and , . Then, and . Thus,
Now, if , then implies . If , then So, we have
It follows that . Therefore, , as needed.
On combining Lemmas 4 and 16, we get the following result.
Theorem 17. A FS of is an -FPI of (1)(2),.
Theorem 18. Every -FPI of is an -FI of .
Proof. Assume that is an -FPI of . Then, ; we have Substitute by in above inequality, so Hence, is an -FI.
Example 19. Take a BCI-algebra of Example 12 defined by Table 2. We define a FS It is easy to calculate that is an -FI of for and ; however, it is not an -FPI as .
Theorem 20. If is an -FPI, then (1), (2),
Proof. (1)Assume that is an -FPI of . So,Substitute for and for , so (2)Since for any . So, by Theorem 18 and Lemma 5, we haveFrom (a), . Thus,
A condition for -FI to be -FPI is given in the following two results.
Theorem 21. Let be an -FI of satisfying Then, is an “-FPI” of .
Proof. Suppose that is an -FI satisfying (15). Then, , following hold Hence, is an -FPI of .
Theorem 22. Let be an -FI of satisfying Then, is an -FPI of .
Proof. Let . Then, by ((17)) and Theorem 20Hence from Theorem 21, is an -FPI of .
Theorem 23. A FS is an “-FPI” in ; the set is a p-ideal of , .
Proof. () Take such that . From Theorem 17, we have
with . Thus, . Therefore, .
Next, assume that and . Then, and . Again, by Theorem 17, we have
So, . Consequently, is p-ideal of .
() Suppose that is a p-ideal of , . If for some , . Then “such that” . It implies that but , a contradiction. So, . Also, if for some , . Then s.t.
It implies that and but , another contradiction. Hence, , as required.
Definition 24. Let be a FS of . The set is said to be an -level subset of .
Theorem 25. A FS of is an -FPI of the -level subset of is a p-ideal of , .
Proof. () Assume that is an -FPI of . Take any . Then, . So, or . By Theorem 17, . Thus when . If , then yields . Also, if , then implies . Similary when .
Next, take any and . Then, and , i.e., either or and either or . By assumption, . We have cases:
Case (i). Let and . If , then
and so, . If , then
So . Hence, .
Case (ii). Let and . If , then
i.e., and, thus, . If . then
and so . Hence, .
Likewise, in the other two cases, i.e., when , and , implying that . Hence, in each case, , and thus .
() Let be “p-ideal” for all Contrary suppose that
with . Then, “such that” . It yields that but , which contradicts itself. Therefore
Also, if for some . Then, such that
Henceforth, and but , another contradiction. Therefore, , as required.
Data Availability
No underlying data was collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2023R87), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.