Abstract
The concept of -BE-algebra is introduced as a generalization of a -BE-algebra and a -BE-algebra and its properties are studied. We introduced the concepts of -subalgebra and -filter of a -BE-algebra and discussed the relation between these two concepts. We provided conditions for an -subalgebra to be an -filter. We provided equivalent conditions for the formation of an -filter from a nonempty subset of a -BE-algebra. We introduced the concept of -atomic -BE-algebra and studied its properties. We introduced the concept of atomic -filter a -BE-algebra and gave the necessary and sufficient condition for an -filter to be atomic.
1. Introduction
The concept of BE-algebras was introduced by Kim and Kim as a generalization of BCK-algebras (see [1]). Later, several authors introduced and studied several substructures of BE-algebras (see [2–4]). In 1964, Nobusawa studied -rings (see [5]). Later, in the year 1966, Barnes weakened the defining conditions of -ring and studied it (see [6]). The structure of -rings was also studied by Ravisankar and Shukla (see [7]). Sen and Saha studied the structure of -semigroups and obtained various generalizations and analogues of corresponding results in the semigroup theory (see [8]). Later, Rao introduced the concept of -semiring as a generalization of -ring and studied various types of -semirings (see [9, 10]). Most algebraic structures are studied in environments where only one binary operation is given, and so is BE-algebra. However, it can be seen that -semigroup, -semiring, and -ring are being studied as algebraic structures with two or more binary operations and these are regarded as generalizations of semigroup, semiring, and ring, respectively. With this motivation, Jun et al. introduced the concept of -BE-algebra as a generalization of a BE-algebra and investigated its properties (see [11]).
In this paper, we introduced the concept of -BE-algebra as a generalization of a -BE-algebra and a -BE-algebra and studied its properties. We introduced the concepts of -subalgebra and -filter of a -BE-algebra and discussed the relation between these two. We provided conditions for an -subalgebra to be an -filter. We provided equivalent conditions for the formation of an -filter from a nonempty subset of a -BE-algebra. We defined and explored the properties of -atomic -BE-algebra. We presented the idea of an atomic -filter and provided a necessary and sufficient condition for an atomic -filter in the context of the -BE-algebra.
2. Preliminaries
A BE-algebra, denoted by (see [1]), is defined to be a set together with a binary operation and a special element, called the unit, satisfying the following conditions: (BE1) (BE2) (BE3) (BE4)
Every BE-algebra meets the following requirements (see [1]):
A subset of a BE-algebra is called as follows:(1) a BE-subalgebra of if it satisfies the following equation:(2) a BE-filter of (see [1]) if it satisfies the following equations:
Let be a nonempty set and let be a set of binary operations on , that is, every element is given as follows:
In what follows, is denoted by , and let be a structure related to a special element and .
Definition 1 (See [11]). For a fixed , a structure is called a -BE-algebra if it satisfies the following equations:If is a -BE-algebra for all , we say that is a -BE-algebra.
3. -BE-Algebras
Definition 2. Given , a structure is called a -BE-algebra if it satisfies the following equations:It is clear that -BE-algebra and -BE-algebra are congruent concepts.
For a fixed , we define a binary relation on as follows:If is valid for all , we present it as .
Example 1. Let be a set and be a set of binary operations on given in Table 1. It is a routine to verify that is a -BE-algebra. But, it is neither a -BE-algebra nor -BE-algebra since and . So, is not even a -BE-algebra.
It is clear that every -BE-algebra is both a -BE-algebra and a -BE-algebra. The following example shows that any -BE-algebra or -BE-algebra may not be a -BE-algebra.
Example 2. (i)Let be a set and be a set of binary operations on given in Table 2. Then, is a -BE-algebra (see [11]). But it is neither a -BE-algebra nor a -BE-algebra since and .(ii)Let be a set and be a set of binary operations on given in Table 3. Then, is a -BE-algebra. But it is neither a -BE-algebra nor a -BE-algebra since and .
Proposition 1. Every -BE-algebra satisfies the following equations:
Proof. Let and . The combination of equations (11), (12), and (14) induces and . Hence, equation (16) is valid. It is clear that equation (17) is true. Using equations (11) and (14), we have and . Thus, and . Equations (19) and (20) are induced from equations (14) and (13), respectively. If , then and so by equations (12) and (14). Hence, . The conditions (22) and (23) are induced by the combination of equations (11), (12), and (14). Let be such that and . If , then by equations (11) and (14), which is a contradiction. Hence, . Let be such that and . If , then by equations (11) and (14). This is a contradiction, and so . Let be such that and . Then, , which proves equation (26). Let be such that and . Then, . Hence, . Now, . Therefore, . Similarly, , that is, . Let be such that and . Then, . Now, . Therefore, . Similarly, . Therefore, .
In general, a -BE-algebra does not satisfy the following assertions:as seen in the example below.
Example 3. The -BE-algebra in Example 1 does not satisfy equation (29) since but , and but . Let be a set and be a set of binary operations on given in Table 4.
It is a routine to verify that is a -BE-algebra. But it does not satisfy equation (30) sinceand
Definition 3. A -BE-algebra is said to be strong if it fulfils condition (29).
Example 4. Let be a set and be a set of binary operations on given in Table 5. It is a routine to verify that is a strong -BE-algebra.
Proposition 2. Every strong -BE-algebra satisfies condition (30).
Proof. It is induced by the combination of equation (18) and (29).
Proposition 3. Every strong -BE-algebra satisfies the following equation:
Proof. Let be such that and . Then, in equation (27), and so by equation (29).
The example below shows that if is not strong, then the condition (33) does not hold.
Example 5. Let be a set and be a set of binary operations on given in Table 6.
It is routine to verify that is a -BE-algebra which is not strong. We can observe that and , but .
4. -Subalgebras and -Filters
In this section, unless otherwise stated, assume that is a -BE-algebra.
Definition 4. A subset of is called an -subalgebra of if it satisfies the following equation:It is clear that for every -subalgebra of .
Example 6. In Example 1, we can observe that is a -subalgebra of .
The following example shows that any -subalgebra of does not satisfy the following assertion:
Example 7. Let be a set and be a set of binary operations on given in Table 7. It is a routine to verify that is a -BE-algebra. Then, is an -subalgebra of , but it does not satisfy equation (35) since but .
Definition 5. If an -subalgebra of satisfies condition (35), we say is a -subalgebra of .
Example 8. Let be a set and be a set of binary operations on given in Table 8.
It is a routine to verify that is a -BE-algebra, and is a -subalgebra of .
It is easy to check that the intersection of -subalgebras is an -subalgebra. But the union of -subalgebras is not an -subalgebra in general as seen in the following example.
Example 9. Let be a set and be a set of binary operations on given in Table 9. It is routine to verify that is a -BE-algebra. Let and . Then, and are -subalgebras of . But is not an -subalgebra of since .
Definition 6. A subset of is called an -filter of if it satisfies equation (4) and the following equations:
Example 10. Let be a set and be a set of binary operations on given in Table 10. It is a routine to verify that is a -BE-algebra. Then, is an -filter of .
The following example shows that any -filter of does not satisfy the following assertion:
Example 11. Let be a set and be a set of binary operations on given in Table 11. It is a routine to verify that is a -BE-algebra. Then, is an -filter of . But does not satisfy equation (38) since and , but .
Definition 7. If an -subalgebra of satisfies condition (38), we say is a -filter of .
Example 12. Let be a set and be a set of binary operations on given in Table 12.
It is a routine to verify that is a -BE-algebra and is a -filter of .
Theorem 1. If is strong, then two conditions (36) and (37) are equivalent to each other for every subset of which contains .
Proof. Let be a subset of which contains . Assume that equation (36) is valid, and let be such that and . Using Proposition 2 leads to , and so by equation (36). Using equation (36) again leads to . Similarly, if satisfies equation (37), then equation (36) is valid.
The next example shows that two conditions (36) and (37) are independent of each other if is not strong. That is, if is a -BE-algebra which is not strong, then there is a subset of such that equation (36) is established, but equation (37) is not established, or equation (37) is established, but equation (36) is not established.
Example 13. (i)Let be a -BE-algebra in Example 7. Then, it is not strong and satisfies equation (36). But it does not satisfy equation (37) since and , but .(ii)Let be a -BE-algebra in Example 5. Then, it is not strong and satisfies equation (37). But does not satisfy equation (37) since and , but .
Proposition 4. Every -filter of satisfies the following equations:
Proof. Straightforward.
Proposition 5. Every -filter of satisfies the following equations:
Proof. Let be an -filter of . Then, . Hence, if you use equations (36) and (37), it connects directly to equation (41). Let and . Then, and . It follows from equations (36) and (37) that and .
We discuss the relationship between an -subalgebra and an -filter.
Theorem 2. Every -filter is an -subalgebra.
Proof. Let be an -filter of and let . Since and , it follows from Proposition 5 that . Therefore, is an -subalgebra of .
The converse of Theorem 2 may not be true as seen in the example below.
Example 14. In Example 1, we can observe that is an -subalgebra of . But is not an -filter of since and , but .
We provide a condition for an -subalgebra to be an -filter.
Theorem 3. If an -subalgebra of satisfies the following equation:then is an -filter of .
Proof. Straightforward.
The combination of Theorems 2 and 3 induces the following corollary.
Corollary 1. An -subalgebra of is an -filter of if and only if it satisfies condition (43).
Theorem 4. If every -subalgebra of satisfies condition (40), then it is an -filter of .
Proof. Let be an -subalgebra of that satisfies condition (40). Clearly, . Let be such that , , and . Then, and , which implies from equation (40) that . In light of this, is an -filter of .
We explore the conditions under which we can make an -filter using a nonempty set.
Theorem 5. Given a nonempty subset of , the following are equivalent.(i) is an -filter of .(ii) satisfies equation (39) and the following equation:
Proof. Let be an -filter of . Then, satisfies equation (39) (see Proposition 4). Let and . Then, It follows from equations (14), (36), and (37) that Hence, and by equations (36) and (37).
Conversely, suppose that satisfies (39) and (44). Since and for all , we have by equation (39). Let and be such that and . Using equations (13) and (44) leads to and .
Hence, is an -filter of .
Definition 8. A nonunit element in is called an atom if the following assertion is valid.
Example 15. Let be a set and be a set of binary operations on given in Table 13.
It is a routine to verify that is a -BE-algebra. The element is the only -atom.
Theorem 6. For every nonunit element in , if is an -filter of , then is an -atom in .
Proof. Assume that is an -filter of . Let be such that and . Then, and . It follows from equations (36) and (37) that . Hence, or . Thus, is an -atom in .
The following example shows that the converse of Theorem 6 may not be true, that is, there exists an -atom in for which is not an -filter of .
Example 16. Let be a set and be a set of binary operations on given in Table 14.
It is a routine to verify that is a -BE-algebra. We can observe that is an atom. But is not an -filter of since , and , but .
Definition 9. A -BE-algebra is said to be -atomic if every nonunit element of is an -atom in .
Theorem 7. If is a -BE-algebra, then the following assertions are equivalent.(i) is -atomic.(ii) satisfies the following equation:(iii) satisfies the following equation:
Proof. (i) (ii). Assume that is -atomic. Let be such that . Note that and . Since is an -atom, we have or and or . If , then and so or . This is a contradiction. Similarly, induces a contradiction. Hence, .(ii) (iii). Straightforward.(iii) (i). Suppose that satisfies (4.14). Let be an arbitrary nonunit element of such that and for all nonunit element . Then, and . If , then and by equations (13) and (51). This is a contradiction, and so , which shows that is an -atom in . Consequently, is -atomic.
Theorem 8. In an -atomic -BE-algebra, every -subalgebra is an -filter.
Proof. Let be an -atomic -BE-algebra. Let be an -subalgebra of . Clearly, . Let be such that , , and . It is clear that if or , then . So we may assume that and are nonunit. Note that and . Since is an -atom, we have or and or . If and , then and . Since is an -atom, it follows that or . Therefore, is an -filter of .
Given a subset of , we consider a set as follows:Note that is the set of all -atoms in . It is clear that and .
Theorem 9. If is an -subalgebra of , then the set is also an -subalgebra of .
Proof. Assume that is an -subalgebra of . Let . It is clear that if , , or , then , . Assume that . Since and , we have or , and or . If or , then or . Since , it follows that , a contradiction. Hence, and , and so and . Therefore, is an -subalgebra of .
Corollary 2. The set is an -subalgebra of .
Question 1. If is an -filter of , then the set is an -filter of ?
The following example provides a negative answer to Question 1.
Example 17. Let be a set and be a set of binary operations on given in Table 15. It is a routine to verify that is a -BE-algebra. We can observe that is an -filter of . But, , and it is not an -filter of since and , but .
Definition 10. An -filter of is said to be atomic if .
Example 18. Let be a set and be a set of binary operations on given in Table 16. It is routine to verify that is a -BE-algebra. Let . Then, is an -filter of and . Hence, is an atomic -filter of .
Theorem 10. In an -atomic -BE-algebra, every -filter is atomic.
Proof. Assume that is an -filter of an -atomic -BE-algebra. It is clear that . Let . Then, is an -atom since is -atomic. Hence, , and so . Therefore, is atomic.
Theorem 11. Let be an -filter of . Then, is atomic if and only if it satisfies the following equation:
Proof. Assume that is an atomic -filter of . Let be such that . If , then and so by equation (13). If , then and so by equation (12). Suppose that . Since and , we have or and or . If or , then or . Since and , it follows that , a contradiction. Hence, and , and thus .
Conversely, suppose that satisfies condition (53). It is clear that . Let be a nonunit element of such that and for all nonunit elements . Then, . If or , then or by condition (53). This is a contradiction, and so and . Thus, , and hence, . Therefore, is an atomic -filter of .
5. Conclusion
We have introduced the concept of -BE-algebra as a generalization of a -BE-algebra and a -BE-algebra and studied its properties. We have introduced the concepts of -subalgebra and -filter of a -BE-algebra and discuss the relation between these two concepts. We have provided conditions for an -subalgebra to be an -filter. We have provided equivalent conditions for the formation of an -filter from a nonempty subset of a -BE-algebra. We have introduced the concept of -atomic -BE-algebra and studied its properties. We have introduced the concept of atomic -filter a -BE-algebra and given a necessary and sufficient condition for an -filter to be atomic.
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia, for funding this research under Researchers Supporting Project number PNURSP2023R231.