Abstract
The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained.
1. Introduction
In 1966, Imai and Iséki [1] introduced the notion of a BCK-algebra. There exist several generalizations of BCK-algebras, such as BCI-algebras [2], BCH-algebras [3], BCC-algebras [4], BH-algebras [5], and d-algebras [6]. Neggers et al. defined B/BM/BG-algebras [7–9] and showed that the class of all B-algebras is a proper subclass of the class of all BG-algebras. They also proved that an algebra is a BM-algebra if and only if it is a 0-commutative B-algebra (therefore, every BM-algebra is a B-algebra). In [10], it is shown that the class of 0-commutative B-algebras is the class of -semisimple BCI-algebras (and hence any BM-algebra is a BCI-algebra). The class of BM-algebras contains Coxeter algebras (see [8]). Some other connections between BM-algebras and its related topics are studied in [11]. Walendziak introduced in [12] the concept of BF-algebras, which is a generalization of B-algebras and BN-algebras defined by C. B. Kim and H. S. Kim [13]. An interesting result of [13] states that an algebra is a BN-algebra if and only if it is a 0-commutative BF-algebra.
We will denote by (resp., ) the class of all BCK-algebras (resp., BCI/BCH/BH/B/BM/BG/BF/BN-algebras). The interrelationships between some classes of algebras mentioned before are visualized in Figure 1 (an arrow indicates proper inclusion; that is, if and are classes of algebras, then means ).
In this paper we consider ideals and fuzzy ideals in BN-algebras. In Section 2, similarly to BCK/BCI/BCH/BH/BF-algebras (see [5, 12, 14–16]), we define the concept of an ideal. We also introduce the notions of normal ideals and normal congruences. We investigate the properties of them and prove that there is a one-to-one correspondence between normal ideals and normal congruences of a BN-algebra. Moreover we obtain the isomorphism theorem for BN-algebras. In this section we also give all what is necessary, to make the paper self-contained. In Section 3 we define fuzzy ideals in BN-algebras (fuzzy ideals of BCK/BCI/BCC/BF-algebras are considered in [17–20]) and provide conditions for a fuzzy set to be a fuzzy ideal. Given a fuzzy set , we make the least fuzzy ideal containing . This leads us to show that the set of fuzzy ideals of a BN-algebra is a complete lattice. Moreover, the homomorphic properties of fuzzy ideals are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras in terms of fuzzy ideals are given in Section 4. Noetherian BCK/BCI/BM/BF-algebras are studied in [21–26].
2. On BN-Algebras: Ideals
An algebra of type is called a Coxeter algebra [27] if for all the following identities hold:,,.
Remark 1. Kim et al. [27] showed that a Coxeter algebra is equivalent to an abelian group all of whose elements have order 2. Therefore, if is a Coxeter algebra, then the operation is commutative and associative.
We say that is a BM-algebra [8] if it satisfies (B2) and.
Proposition 2 (see [8]). If is a BM-algebra, then, for any ,(i),(ii).
Proposition 3. If is a BM-algebra, then, for all ,(i),(ii)if , then .
Proof. (i) Substituting and in (BM), we have . Applying (B2) we get (i).
(ii) follows from (i) and (B2).
An algebra of type is called a BF-algebra [12] if it satisfies (B1), (B2), and.
Definition 4. An algebra of type is called a BN-algebra [13] if (B1), (B2), andhold for all .
Let be a BN-algebra. We define a binary relation on by if and only if . It is easy to see that, for any , if , then .
Example 5. Let be the set of real numbers and let be the algebra with the operation defined by Then is a BN-algebra.
Example 6 (see [13]). Let and be defined by the following: Then is a BN-algebra.
Example 7. Let be an abelian group. If we define , for all , then, by Theorem 2.15 of [13], is a BN-algebra.
Proposition 8 (see [13]). If is a BN-algebra, then, for all , (i),(ii),(iii),(iv),(v).
From [13] we obtain the following relation: .
From now on, always denotes a BN-algebra . We introduce the notion of an ideal in BN-algebras.
Definition 9. A subset of is called an ideal of if it satisfies , and imply for any .
We will denote by the set of all ideals of a BN-algebra .
Example 10. Let be the BN-algebra given in Example 5. Observe that . Indeed, let be an ideal of and suppose that . Then for some . Let . Obviously, and hence . Therefore, .
Proposition 11. Let and . If and , then .
Proof. Let and . Then and . Hence .
A nonempty subset of is called a subalgebra of if for any . It is easy to see that if is a subalgebra of , then .
Example 12. Let , where is the set of all integers. We know that is a BN-algebra (see Example 7). Obviously, a nonempty subset is a subalgebra of if and only if for all ; that is, is a subgroup of the group of integers. Therefore,
Remark 13. Consider the BN-algebra given in Example 6. We have . Observe that the ideal is not a subalgebra of and is a subalgebra of but it is not an ideal.
Remark 14. It is easy to prove that the intersection of an arbitrary number of ideals of a BN-algebra is an ideal of . It is also not hard to show that the union of an ascending sequence of ideals of is an ideal of .
A nonempty subset of is said to be normal in if for any . We say that an ideal of (resp., a subalgebra of ) is normal if the set (resp., the set ) is normal. We will denote by the set of all normal ideals of .
Remark 15. It is easy to see that , . The ideal is normal, but is not normal in general (see the example below).
Example 16. Consider the set with the operation defined by the following: It is easily seen that is a BN-algebra. The ideal is not normal because , but .
Proposition 17. If is a normal ideal of , then is a subalgebra of .
Proof. Let . Since is normal, we conclude that ; that is, . From the definition of an ideal we have . Thus is a subalgebra of .
Remark 18. The BN-algebra given in Example 16 shows that the converse of Proposition 17 does not hold. Indeed, is a subalgebra of but it is not a normal ideal.
Proposition 19. Every ideal of a Coxeter algebra is normal.
Proof. Let be a Coxeter algebra and let be an ideal of . Let and suppose that . Since is a subalgebra, we have . From Remark 1 it follows that . Thus is normal.
Lemma 20. Let be a normal ideal of a BN-algebra and . Then (a),(b).
Proof. (a) Let . Then . Since is normal, . Hence, .
(b) Let . By (a), . Applying Proposition 8(ii) we have .
Proposition 21. Let be a BN-algebra and let . Then is a normal subalgebra of if and only if is a normal ideal.
Proof. Let be a normal subalgebra of . Clearly, . Suppose that and . Then . Since is normal, we have . By (B1) and (B2), . Therefore , and thus is an ideal. The converse follows from Proposition 17.
Remark 22. Let be a normal ideal of a BN-algebra . For any , we define Then is a congruence of by the proof of Theorem 3.5 of [13].
Definition 23. Let be a congruence of a BN-algebra . One says that is normal if for arbitrary .
Example 24. Let and be defined by the following: It is easy to check that is a BN-algebra and is a normal congruence of .
By we denote the set of all normal congruences of .
Remark 25. Let be a BN-algebra. It is easily seen that , where is a normal ideal of . In particular, .
Proposition 26. In BM-algebras all congruences are normal.
Proof. Let be a BM-algebra and let be a congruence of . Let and suppose that and . Hence, . Observe that Indeed, we have Therefore, and thus is normal.
Remark 27. Let be the BN-algebra from Example 16. Observe that the least congruence on , namely, the identity relation , is not normal. Indeed, we get and but .
Let be a BN-algebra and let be a congruence on . For , by we denote the congruence class containing ; that is, .
Proposition 28. Let be a congruence on . Then if and only if is a normal ideal of .
Proof. Set and let . It follows easily that . Let and be elements of such that . Then, and . Since , we obtain and hence . Consequently, is an ideal. Now, suppose that , where . Then, and . By the definition of a normal congruence, ; that is, . Thus, is normal.
The converse is obvious.
Theorem 29. There is a bijection between the set of normal ideals and the set of normal congruences of a BN-algebra.
Proof. Let be a BN-algebra. We consider functions and given as follows: Since and , we conclude that and are well-defined. Next, note that . Indeed, Hence, Further, observe that and from this we obtain We deduce from 13 and 15 that and . Thus, and are inverse bijections between and .
Let and be BN-algebras. A mapping is called a homomorphism from into if for any .
Observe that . Indeed, . We denote by the subset of (it is the kernel of the homomorphism ).
Lemma 30. Let be a homomorphism from into . Then is an ideal of .
Proof. Obviously, ; that is, () holds. Let and . Then . Consequently, . Therefore, () is satisfied. Thus is an ideal of .
Remark 31. The kernel of a homomorphism is not always a normal ideal. Let be the algebra given in Example 16. Clearly, is a homomorphism and the ideal is not normal.
Proposition 32. Let be a BN-algebra and let be a BM-algebra. Let be a homomorphism from into . Then is a normal ideal.
Proof. By Lemma 30, is an ideal of . Let and . Then . From Proposition 3(ii) it follows that . Similarly, . Consequently, , and hence, .
Let be a normal ideal of . For , we write ; that is, . We note that Denote and set . The operation is well-defined, since is a congruence of . It is easy to see that is a BN-algebra. The algebra is called the quotient BN-algebra of modulo .
Example 33. Let and be given as in Example 24. We know that is a BN-algebra and is a normal congruence of . Since , from Proposition 28 we see that is a normal ideal of . We have and . Then . Clearly, and .
Theorem 34. Let be a BN-algebra and let be a BM-algebra. Let be a homomorphism from onto . Then is isomorphic to .
Proof. By Proposition 32, is a normal ideal of . Define a mapping by for all . Let . Then, ; that is, . Hence, . By Proposition 3(ii) we have . Consequently, . This means that is well-defined. It is easy to check that is a homomorphism from onto . Observe that is one-to-one. Let . Then and hence ; that is, . Therefore, and consequently, . Thus is an isomorphism from onto .
3. Fuzzy Ideals
We now review some fuzzy logic concepts. First, for we define and . Obviously, if , then and . Recall that a fuzzy set in is a function .
For any fuzzy sets and in , we define A trivial verification shows that this relation is an order relation in the set of fuzzy sets in .
Let and be any two sets, any fuzzy set in , and any function. Set for . The fuzzy set in defined by for all , is called the image of under and is denoted by .
Let and be any two sets, any function, and any fuzzy set in . The fuzzy set in defined by is called the preimage of under and is denoted by .
Now, we give the definition of a fuzzy ideal in a BN-algebra.
Definition 35. A fuzzy set in is called a fuzzy ideal of a BN-algebra if it satisfies, for all , , .
Proposition 36. Let be a fuzzy ideal of . Then, for any , if , then .
Proof. If , then . From Proposition 8(iv) we obtain . Hence, by (d2) and (d1), we have .
Denote by Id the set of all fuzzy ideals of a BN-algebra .
Example 37. Let be the BN-algebra given in Example 6. Let . Define a fuzzy set in by It is easily checked that satisfies (d1) and (d2). Thus .
Example 38. Let be an ideal of a BN-algebra and let with . Define as follows:
We denote . Since , for all . To prove (d2), let . If , then . Suppose now that . By the definition of an ideal, or . Therefore, . Thus is a fuzzy ideal of .
In particular, the characteristic function of :
is a fuzzy ideal of .
Proposition 39. A fuzzy set in is a fuzzy ideal of if and only if it satisfies and for all , if , then .
Proof. Let and let . Suppose that . Since is a fuzzy ideal, we have and . Therefore, .
Conversely, let satisfy (d3). From (B1) we have . By (d3), . Then satisfies (d2) and hence .
Theorem 40. Let be a fuzzy set in . Then if and only if its nonempty level subset is an ideal of for all .
Proof. Assume that . Let and . Then for some . Since , we have . Let such that . Then and . It follows from (d2) that
so that . Therefore is an ideal of .
Conversely, suppose that, for each or is an ideal of . If (d1) is not valid, then there exists such that . Then and, by assumption, is an ideal of . Hence and consequently . This is a contradiction and (d1) is valid. Now assume that (d2) does not hold. Then there are such that . Taking
we get and . Therefore but . This is impossible, and is a fuzzy ideal of .
By Theorem 40, we have the following.
Corollary 41. If is a fuzzy ideal of a BN-algebra , then the set is an ideal of .
The following example shows that the converse of Corollary 41 does not hold.
Example 42. Let be a BN-algebra. Define a fuzzy set in by Then is the ideal of but (because does not satisfy (d1)).
Example 43. Consider the BN-algebra given in Example 6. Let be defined as in Example 37. It is easy to check that for all we have Since and are ideals of , this is another proof (by Theorem 40) that is a fuzzy ideal of .
Lemma 44. Let be a strictly ascending sequence of ideals in a BN-algebra and let be a strictly decreasing sequence in . Let be the fuzzy set in defined by where . Then is a fuzzy ideal of .
Proof. Let . By Remark 14, is an ideal of . Obviously, for all ; that is, (d1) holds. Now we show that satisfies (d2). Let . We have two cases.
Case 1 . Then or . Therefore .
Case 2 for some . Then or . Hence or . Therefore .
Thus (d2) is also satisfied and consequently is a fuzzy ideal of .
Let be a nonempty set of indexes. Let for . The meet of fuzzy ideals of is defined as follows:
Theorem 45. Let for . Then .
Proof. Let . Then, by (d1), for all . Let . Since , we have . Hence Consequently, and therefore .
Let be a fuzzy set in . A fuzzy ideal of is said to be generated by if and, for any fuzzy ideal of , implies . The fuzzy ideal generated by will be denoted by . The fuzzy ideal can be defined equivalently as follows:
For let denote the join of and ; that is, , where is the fuzzy set in defined by for all .
From Theorem 45 we obtain the following theorem.
Theorem 46. Let be a BN-algebra. Then is a complete lattice.
The following two theorems give the homomorphic properties of fuzzy ideals.
Theorem 47. Let and be BN-algebras and let be a homomorphism and . Then .
Proof. Let . Since and , we have , but . Thus we get for any ; that is, satisfies (d1).
Now let . Since , we obtain
and hence . Consequently, .
Lemma 48. Let and be BN-algebras and let be a homomorphism and . Then, if is constant on , then .
Proof. Let and . Hence For all , we have . Hence ; that is, . Thus . Therefore, . Similarly, . Hence . Thus that is, .
Theorem 49. Let and be BN-algebras and let be a surjective homomorphism and such that . Then .
Proof. Since is a fuzzy ideal of and , we have for any . Hence for any . Thus satisfies (d1). Suppose that for some . Since is surjective, there are such that and . Hence Therefore Since , is constant on . Then, by Lemma 48, we get which is a contradiction with the fact that is a fuzzy ideal. Thus, we obtain .
4. Fuzzy Characterizations of Noetherian and Artinian BN-Algebras
In this section we characterize Noetherian BN-algebras and Artinian BN-algebras using some fuzzy concepts, in particular, fuzzy ideals.
A BN-algebra is called Noetherian if for every ascending sequence of ideals of there exists such that for all . A BN-algebra is called Artinian if for every descending sequence of ideals of there exists such that for all .
Theorem 50. Let be a BN-algebra. The following statements are equivalent: (a) is Noetherian,(b)for each fuzzy ideal of , is a well-ordered set.
Proof. (a) (b): Assume that is Noetherian and is a fuzzy ideal of such that is not a well-ordered subset of . Then there exists a strictly decreasing sequence , where . Let and . Then, by Theorem 40, is an ideal of for every . So is a strictly ascending sequence of ideals of . This is a contradiction with the assumption that is Noetherian. Therefore is a well-ordered set for each fuzzy ideal of .
(b) (a): Assume that (b) is true. Suppose that is not Noetherian. Then there exists a strictly ascending sequence of ideals of . Let be a fuzzy set in such that
where . By Lemma 44, , but is not a well-ordered set, which is impossible. Therefore is Noetherian.
Corollary 51. Let be a BN-algebra. If, for every fuzzy ideal of , is a finite set, then is Noetherian.
Theorem 52. Let be a BN-algebra and let , where is a strictly decreasing sequence in . Then the following conditions are equivalent: (a) is Noetherian,(b)for each fuzzy ideal of , if , then there exists such that .
Proof. (a) (b): Assume that is Noetherian. Let be a fuzzy ideal of such that . From Theorem 50 we know that is a well-ordered subset of . Then, since and , there exists such that .
(b) (a): Assume that (b) is true. Suppose that is not Noetherian. Then there exists a strictly ascending sequence of ideals of . Define a fuzzy set in by
where . By Lemma 44, is a fuzzy ideal of . This is a contradiction with our assumption. Thus is Noetherian.
Theorem 53. Let be a BN-algebra and let , where is a strictly increasing sequence in . Then the following conditions are equivalent: (a) is Artinian,(b)for each fuzzy ideal of , if , then there exists such that .
Proof. (a) (b): Suppose that is a strictly increasing sequence of elements of . Let for . It is immediately seen that is a strictly descending sequence of ideals of . This contradicts the assumption that is Artinian.
(b) (a): Assume that (b) is true. Suppose that is not Artinian. Then there exists a strictly descending sequence of ideals of . Define a fuzzy set in by
Obviously, for all ; that is, (d1) holds. Now we show that satisfies (d2). Let . We have three cases.
Case 1 . Then or . Therefore .
Case 2 for some . Then or . Hence or . Therefore .
Case 3 . It is obvious.
Thus is a fuzzy ideal of . This contradicts our assumption. Thus is Artinian.
Corollary 54. Let be a BN-algebra. If, for every fuzzy ideal of , is a finite set, then is Artinian.
The following example shows that the converse of Corollary 54 does not hold.
Example 55. Let be a prime number. Set for some . It is known that is the -quasicyclic group. Define for all . By Example 7, is a BN-algebra. Let for . It follows easily that is an ideal of if and only if or for some . We have and hence is Artinian. Define by where . Since , is a fuzzy set in . By the proof of Lemma 44, . However, is not a finite set.
5. Conclusions
This paper begins by considering the notion of ideals in BN-algebras. We give its characterizations and introduce the concept of normal ideals investigating its properties. We also define the notion of normal congruences proving that there is a one-to-one correspondence between normal ideals and normal congruences of a BN-algebra. Moreover we obtain the isomorphism theorem for BN-algebras. Next, we define the notion of fuzzy ideals of BN-algebras giving its characterizations and providing conditions for a fuzzy set to be a fuzzy ideal. We give the relationships between ideals and fuzzy ideals of a BN-algebra and also provide the homomorphic properties of fuzzy ideals. Finally, we display characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals.
The next step in studying fuzzy ideals in BN-algebras may be introducing and investigating the notions of fuzzy maximal ideals and fuzzy prime ideals of BN-algebras.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the referees for remarks which were incorporated into this revised version.