1. Introduction
The notion of BCK-algebras was initially formulated by Imai and Iséki in 1966 [
1]. These algebraic structures, named after Blokh, Cohn, and Kisielewicz, have since become essential tools in the intersection of logic and algebra. In contrast with conventional Boolean algebras, BCK-algebras provide a more adaptable framework for logical operations, particularly by emphasizing implications and relationships in a manner that broadens the scope of classical algebraic systems. The exploration of BCK-algebras opens new possibilities in theoretical mathematics and computer science, offering valuable insights into logical systems and algorithmic processes.
The Sheffer operation, or Sheffer stroke (denoted as “∣”), was introduced by H. M. Sheffer [
2]. This operation is noteworthy because it can independently form a complete logical system, meaning that all axioms within such a system can be expressed solely through the Sheffer stroke. This self-sufficiency simplifies the management and manipulation of logical systems. Furthermore, Boolean algebra, which forms the algebraic foundation of classical propositional logic, can also be entirely represented through the Sheffer operation. Its functional completeness enables the construction of other logical operations such as AND, OR, and NOT, making it a fundamental element in both Boolean algebra and digital circuit design. The Sheffer operation is particularly recognized for streamlining algebraic structures by reducing the number of axioms, thus making the study of systems incorporating this operation more manageable. Prominent examples include Boolean algebras [
3] and ortholattices [
4]. More recently, the Sheffer operation has found applications in a variety of algebraic structures, such as basic algebras [
5], Sheffer stroke Hilbert algebras [
6], and UP-algebras [
7], as outlined in the existing literature. This broad range of applications demonstrates the Sheffer operation’s versatility and its capacity to simplify and advance the study of complex algebraic systems.
BCK-algebras, originally proposed by Imai and Iséki, draw inspiration from set theory and both classical and non-classical propositional calculi. These algebras have seen widespread use in areas such as group theory, functional analysis, probability theory, and topology. A comprehensive study in [
8] investigates the combination of BCK-algebras with the Sheffer operation, providing an in-depth exploration of their properties and applications in these fields.
Bipolar fuzzy sets generalize classical fuzzy set theory by assigning to each element a pair of membership values, representing both its degree of positive and negative association. This duality allows for a more sophisticated approach to modeling uncertainty, particularly in cases where both positive and negative information are relevant. Unlike classical fuzzy set theory, where each element has a single degree of membership indicating truth or belonging, bipolar fuzzy sets recognize that real-world situations often require consideration of both favorable and unfavorable degrees simultaneously. This concept has been instrumental in advancing fields such as decision-making, pattern recognition, and information processing, where ambiguity and contradictions are prevalent. Studying bipolar fuzzy sets within various algebraic structures, like BCK-algebras, creates new opportunities for understanding their properties and applications, yielding deeper insights into the behavior of logic systems under uncertain conditions.
This study investigates bipolar fuzzy SBCK-subalgebras and the level sets of bipolar fuzzy sets within the framework of Sheffer stroke BCK-algebras. These constructs are instrumental in capturing the behavior of bipolar logic in the context of this algebraic structure. A central result of the paper is the establishment of a bidirectional correspondence between bipolar fuzzy SBCK-subalgebras and their associated level sets. It is demonstrated that each level set derived from a bipolar fuzzy SBCK-subalgebra forms a subalgebra, and, conversely, each such subalgebra determines a level set of an appropriate bipolar fuzzy set.
Moreover, the concept of a bipolar fuzzy SBCK-ideal is introduced and its algebraic properties are thoroughly examined. It is proven that any bipolar fuzzy SBCK-ideal inherently qualifies as a bipolar fuzzy SBCK-subalgebra. However, the converse does not generally hold, indicating that bipolar fuzzy SBCK-ideals possess distinctive structural features that differentiate them from more general subalgebras within Sheffer stroke BCK-algebras.
The list of acronyms is given in
Table 1.
4. Bipolar Fuzzy Sets in SBCK-Ideals
This section investigates bipolar fuzzy SBCK-ideals within the context of SBCK-algebras. The theorems presented establish a formal framework for identifying the conditions under which these fuzzy sets qualify as SBCK-ideals, thereby enhancing their applicability in logical and algebraic contexts.
Definition 8. A BFS on is called a of if Example 3. Let be an SBCK-algebra given in Example 1. Define a BVFS in by the table below: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| | | | | | | | |
| | | | | | | | |
It is routine to verify that the BFS in is a of .
Lemma 3. If is a BFI of , then Proof. Let
be a BFI of
and
. Then, by Theorem 2, we have
for all
. □
Theorem 4. Let be a BFI of X. Then, is a BFI of , if and only if Proof. Let
be a BFI of
, and assume that
. Then,
Furthermore, we have
and similarly
From this, we deduce that
and
Conversely, suppose that
is a BFS of
satisfying the condition in (4). Since
we obtain
Next, as
we conclude that
Moreover,
Thus, we obtain
and
Therefore,
is a BFI of
. □
Theorem 5. Every of is a of .
Proof. Let be a of . Then
and
Hence
is a
of
X. □
Theorem 6. A BFS in is a of if and only if its negative -cut and positive -cut are SBCK-ideals of , whenever they are nonempty for all .
Proof. Assume that
is a
of
and
for all
. Let
be such that
. Then, we have
Using (1), we obtain
Thus,
. Let
be such that
and
. Then, we have
Using (1), we obtain
and
Thus,
. Therefore,
and
are SBCK-ideals of
.
Conversely, let be a BFS in for which its negative -cut and positive -cut are SBCK-ideals of whenever they are nonempty for all .
Suppose that for some . Then, but , a contradiction. Hence, for all .
Suppose that for some . Then, but , a contradiction. Hence, for all .
Suppose that or for some . Then, or where and .
But or , present a contradiction. Therefore, and for all .
Consequently, is a of . □
Theorem 7. A BFS in is a of if and only if the fuzzy sets and are fuzzy SBCK-ideals of , where Proof. Let be a of . It is clear that is a fuzzy SBCK-ideal of .
For every
,
and
Hence,
is a fuzzy SBCK-ideal of
X.
Conversely, let
be a BFS of
, for which
and
are fuzzy SBCK-ideals of
. Let
. Then, as
we obtain
and as
we have
Hence,
is a
of
. □
Theorem 8. Given a nonempty subset of , let be a BFS in , defined as follows:where in and in . Then, is a of , if and only if is an SBCK-ideal of . Proof. Assume that
is a BFS of
. Let
be such that
. Using (1), we have
and so
This shows that
. Next, we have
and so
We have
. Therefore,
is an SBCK-ideal of
.
Conversely, let
be an SBCK-ideal of
. For every
, if
, then
, which implies that
If
, then
For every
, if
, then
, which implies that
and
If
or
, then
and
Therefore,
is a
of
. □
Proposition 3. If is a family of bipolar fuzzy SBCK-ideals of , then is a of .
Proof. Let
be a family of bipolar fuzzy SBCK-ideals of a SBCK-algebra
X. Let
. We have
and
For
, we have the following inequality:
Similarly, we have
Hence,
is a
of
. □
5. Bipolar Fuzzy -Translations in SBCK-Algebras
In this section, we introduce and explore the concept of bipolar fuzzy -translations within SBCK-algebras.
Definition 9. The inclusion ⊆ is defined as follows: for any bipolar fuzzy sets and on ,We say that is a bipolar fuzzy extension of , and is a bipolar fuzzy intension of . Example 4. Let be a SBCK-algebra given in Example 1. Define bipolar-valued fuzzy sets and in X by the table below: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
It is routine to verify that the BFS is a bipolar fuzzy extension of , and is a bipolar fuzzy intension of .
Definition 10. For any BFS in , we denoteLet be a BFS in and . By a of of type I, we mean a BFS , where Lemma 4. If a BFS in is a of , then for all , a of is a of .
Proof. Assume that
is a
of
. For any
and for all
, we have
and
Hence,
is a
of
. □
Lemma 5. If there exists , such that the translation of is a of , then is a of .
Proof. Assume that
is a
of
for
and for all
, we have
and
Hence,
and
. Hence,
is a
of
. □
Lemma 6. If a BFS in is a of , then for all , a of is a of .
Proof. Assume that
is a
of
. For any
and for all
, we have
Then,
Next, let
. Then,
and
Then, we have
and
Hence,
is a
of
. □
Theorem 9. If there exists , such that the BF translation of is a of , then is a of .
Proof. Assume that
is a
of
for
, and for all
, we have
Then, we have the following:
and
Now, let
. Then, we have the following:
and
Hence, we obtain the following inequalities:
and
Therefore,
is a
of
. □
Remark 1. If is a BFS of , then for all , we haveandfor all . Hence, the of is a bipolar fuzzy extension of for all . Definition 11. For any BFS, in , we denoteLet be a BFS in and . By a of of Type II, we mean a BFS , where Theorem 10. If a BFS in is a of , then for all , a BF translation of is a of .
Proof. Assume that
is a
of
. For any
and for all
, we have
Hence,
is a
of
. □
Theorem 11. If there exists , such that the BF translation of is a of , then is a of .
Proof. Assume that
is a
of
for
and for all
, we have
and
Hence, we obtain
and
. Therefore,
is a
of
. □
Theorem 12. If a BFS in is a of , then for all , a BF translation of is a of .
Proof. Assume that
is a
on
. For any
and for all
, we have the following relations:
and
Next, let
. Then, it follows that
Thus, we have
and
Therefore,
is a
of
. □
Theorem 13. If there exists , such that the BF translationof is a of , then is also a of . Proof. Assume that
is a
of
for
and for all
, we have
Then,
and
Now, let
. Then
and
Hence,
and
Therefore,
is a
of
. □
Remark 2. If is a BFS of , then for all ,andfor all . Hence, the ,is a bipolar fuzzy extension of for all . Definition 12. Let be a BFS of . The BFS is defined for all asandThis BFS is called the complement of in . Definition 13. Let be a BFS in . For , the setsandare called the negative lower -cut and the negative upper -cut of , respectively. The setsandare called the positive lower -cut and the positive upper -cut of , respectively. Theorem 14. Let be a BFS in . Then, is a of if and only if for all , and are SBCK-subalgebras of if and are nonempty.
Proof. Assume that is a of . Let be such that and are nonempty.
- (i)
Let
. Then,
and
, so
is a lower bound of
. As
is a
of
, we have
Therefore, , and hence is an SBCK-subalgebra of .
- (ii)
Let
. Then
and
, so
is an upper bound of
. As
is a
of
, we obtain
Thus,
and so
. Therefore,
is an SBCK-subalgebra of
.
Conversely, assume that for all , and are SBCK-subalgebras of if they are nonempty.
- (i)
Let
. Then,
. Choose
. Thus,
and
, so
. By assumption, we have
is an SBCK-subalgebra of
and so
. Thus,
- (ii)
Let
. Then
. Choose
. Thus
and
, so
. By assumption, we have
is an SBCK-subalgebra of
and so
. Thus,
Thus, is a of . □
Theorem 15. Let represent a BFS in . Then, qualifies as a of , if and only if, for every pair , the sets and are SBCK-ideals of , provided that these sets are nonempty.
Proof. Assume that
is a
of
. Let
be such that
and
are nonempty. Let
be such that
. Then,
. As
is a
of
, we have
Hence,
.
Next, let
be such that
. Then
and
. As
is a
of
, we have
Hence,
. Therefore,
is an SBCK-ideal of
.
Let
be such that
. Then,
. As
is a
of
, we have
Hence,
.
Now, let
be such that
. Then,
and
. As
is a
of
, we have
Hence,
. Therefore,
is an SBCK-ideal of
.
Conversely, assume that for all
, the sets
and
are SBCK-ideals of
, provided that these sets are non-empty. Let
. Then, we have
. Choose
. Consequently,
, which implies
. By assumption,
is an SBCK-ideal of
, so
. Thus, we obtain
, and consequently,
Now, let
. Then, we have
. Choose
. Therefore,
and
, implying that
. By assumption,
is an SBCK-ideal of
, so
. Thus,
Now,
Let
. Then, we have
. Choose
. Hence,
, implying
. By assumption,
is an SBCK-ideal of
, so
. Thus, we have
, and consequently,
Let
. Then, we have
. Choose
. Therefore,
and
, implying that
. By assumption,
is an SBCK-ideal of
, so
. Thus,
Now,
Hence,
is a
of
. □
Definition 14 ([
8])
. Let and denote SBCK-algebras. A mapping is called a homomorphism if it satisfies the condition for all , and additionally, . As a notational convenience, we denote
Theorem 16. Let and be SBCK-algebras, and let be a surjective homomorphism. Suppose ℏ is a BFS on . Then, ℏ is a of if and only if is a of , where on is defined by for all .
Proof. Let
and
be SBCK-algebras, with
as a surjective homomorphism and
ℏ being a
of
. Let
. Then
and
Besides, we have
and
Thus, is a of .
Conversely, suppose
is a
of
P. Let
, such that
and
for
. Then, we have
and
Morever, we obtain
and
Hence,
ℏ is a
of
. □
Theorem 17. Let and be SBCK-algebras, and let be a surjective homomorphism. Assume that ℏ is a BFS in . Then, ℏ is a bipolar fuzzy of , if and only if is a of , where on is defined by for all .
Proof. Let
and
be SBCK-algebras, and let
be a surjective homomorphism. Assume that
ℏ is a
of
. Let
. Then
and
Hence,
is a
of
.
Conversely, let
be a
of
. Let
, such that
and
for
. Then
and
Hence,
ℏ is a
of
. □
6. Conclusions
In this paper, we have thoroughly examined the concept of bipolar fuzzy SBCK-subalgebras and their level sets within the framework of Sheffer stroke BCK-algebras. Our primary contribution is the identification of a key relationship between these subalgebras and the corresponding level sets in Sheffer stroke BCK-algebras, specifically proving that the level set of a bipolar fuzzy SBCK-subalgebra corresponds to a subalgebra, and the reverse holds as well. This finding enriches the algebraic structure of Sheffer stroke BCK-algebras and lays a foundational framework that may support further exploration of bipolar fuzzy logic within the context of Sheffer stroke BCK-algebras.
Furthermore, we introduced the notion of a bipolar fuzzy SBCK-ideal, which represents a significant extension of the concept of SBCK-subalgebras in Sheffer stroke BCK-algebras. We demonstrated that while every bipolar fuzzy SBCK-ideal is also a bipolar fuzzy SBCK-subalgebra, the converse does not necessarily hold. This distinction highlights the unique characteristics of bipolar fuzzy SBCK-ideals and their behavior within the broader algebraic structure. These results provide a clearer understanding of the role of fuzzy logic in the theory of BCK-algebras and expand upon previous work in this area.
Despite the valuable contributions of this study, several directions for future research remain. These areas promise to extend the scope and applicability of the concepts discussed in this paper.
Extension to Other Algebraic Structures: While this paper focused on Sheffer stroke BCK-algebras, future work could investigate the applicability of the identified relationships between subalgebras and level sets in other types of algebraic structures, such as general BCK-algebras or other non-classical logical algebras. Understanding whether similar results hold in these broader contexts could provide a more generalized framework for bipolar fuzzy logic.
Interaction with Other Logical Operations: Another natural extension of this work is to study the interaction between bipolar fuzzy SBCK-subalgebras and other logical operations within the context of non-classical logics, such as fuzzy logic, intuitionistic logic, and paraconsistent logic. Investigating how these operations behave in the presence of bipolar fuzzy sets and how they can be integrated with the Sheffer stroke operation could lead to new insights into logical systems and their applications.
Practical Applications in Computational Systems: While this study has been theoretical, the practical applications of bipolar fuzzy SBCK-subalgebras and SBCK-ideals in computational systems are vast. Future research could explore how these algebraic structures can be applied in areas such as fuzzy decision-making, multi-criteria decision analysis, and information retrieval systems, where bipolar fuzzy sets often provide a useful model for uncertain or ambiguous information. Additionally, the study of how these concepts can improve algorithms for machine learning or artificial intelligence, particularly in areas that require the processing of imprecise or conflicting data, could lead to valuable advancements in computational theory.
Algorithmic Approaches: Future research could also focus on the development of efficient algorithms for computing and manipulating bipolar fuzzy SBCK-subalgebras and SBCK-ideals. Designing algorithms that can automatically identify and work with these structures could facilitate their application in real-world systems, particularly in contexts where large amounts of data are involved, such as data mining or complex decision support systems.
Further Investigation of Properties of SBCK-Ideals: A more in-depth study of the properties and behavior of bipolar fuzzy SBCK-ideals within Sheffer stroke BCK-algebras is another promising direction. Specifically, investigating the conditions under which the reverse of the relationship between bipolar fuzzy SBCK-ideals and SBCK-subalgebras might hold could lead to new insights and contribute to a more complete characterization of these structures.
Cross-disciplinary Applications: The theoretical framework established in this paper may also find applications in fields outside of pure mathematics, such as computer science, economics, and cognitive science. By applying bipolar fuzzy logic to model decision-making processes, economic behaviors under uncertainty, or even cognitive reasoning, future research could explore the real-world impact of these algebraic concepts in interdisciplinary domains.
In conclusion, while the results presented in this paper represent a significant step forward in the study of bipolar fuzzy SBCK-subalgebras and their associated level sets in Sheffer stroke BCK-algebras, there are many exciting avenues for further research that could expand upon these findings and open up new directions for both theoretical exploration and practical application.