On Fuzzy Ideals of BL-Algebras

Abstract

In this paper we investigate further properties of fuzzy ideals of a BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime ideal ω is a fuzzy irreducible ideal if and only if and . We give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of a BL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.

1. Introduction

It is well-known that an important task of the artificial intelligence is to make computer simulate human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic includes many valued logic and fuzzy logic which takes the advantage of the classical logic to handle information with various facets of uncertainty [1], such as fuzziness and randomness. Therefore, nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. Fuzziness and incomparability are two kinds of uncertainties often associated with human’s intelligent activities in the real word, and they exist not only in the processed object itself, but also in the course of the object being dealt with.

The notion of BL-algebra was initiated by Hájek [2] in order to provide an algebraic proof of the completeness theorem of Basic Logic (, in short). A well known example of a -algebra is the interval endowed with the structure induced by a continuous -norm. -algebras [3], Gödel algebras, and Product algebras are the most known class of -algebras. Cignoli et al. [4] proved that Hájek’s logic really is the logic of continuous -norms as conjectured by Hájek. Filters theory plays an important role in studying -algebras. From logic point of view, various filters correspond to various sets of provable formulae. Hájek introduced the notions of filters and prime filters in -algebras and proved the completeness of Basic Logic using prime filters. Turunnen [5–7] studied some properties of deductive systems and prime deductive systems. Haveshki et al. [8, 9] introduced (positive, fantastic) implicative filters in -algebras and studied their properties.

The concept of fuzzy sets was introduced by Zadeh [10]. At present, these ideals have been applied to other algebraic structures such as groups and rings. Liu et al. ([11, 12]) introduced the notions of fuzzy filters and fuzzy prime filters in -algebras and investigated some of their properties. Zhan et al. [13–16] introduced some kinds of generalized fuzzy filters in -algebras and described their relations with ordinary fuzzy filters. Another important notion of -algebras is ideal, which was introduced by Hájek [2]. Some properties of ideals were investigated by Saeid [17]. Fuzzy ideal theory in -algebras is studied by Zhang et al. [15]. The notions of fuzzy prime ideals and fuzzy Boolean ideals are introduced.

In the present paper we will systematically investigate fuzzy ideal theory of -algebras. The paper is organized as follows. In Section 2, we recall some basic definitions and results of -algebras. In Section 3, we provide a procedure to generate a fuzzy ideal by a fuzzy set. In Section 4, the notions of fuzzy irreducible ideals and fuzzy Gödel ideals are introduced. We give a new definition of fuzzy prime ideals in a -algebra and prove that it is equivalent to one in Zhang et al. [15]. We prove that every fuzzy irreducible ideal in a -algebra is a fuzzy prime ideal and give an example to show that a fuzzy prime ideal may not be a fuzzy irreducible ideal; also we prove that a fuzzy prime ideal is a fuzzy irreducible ideal if and only if and . Furthermore, we give the Krull-Stone representation theorem of fuzzy ideals in -algebras. In Section 5, we prove that the lattice of all fuzzy ideals of a -algebra is a complete distributive lattice. Finally, in Section 6, we introduce the notion of fuzzy Gödel ideals and investigate basic properties of fuzzy Gödel ideals and prove that every fuzzy Boolean ideal is a fuzzy Gödel ideal but the converse implication is not true.

2. Preliminaries

Let us recall some definitions and results on -algebras.

Definition 1 (see [2]). An algebra of type is called a -algebra if it satisfies the following conditions.(BL1) is a bounded lattice.(BL2) is a commutative monoid.(BL3) if and only if (residuation).(BL4); thus (divisibility).(BL5) (prelinearity).
Throughout this paper, let denote a -algebra.

Proposition 2 (see [5, 9]). Let be a -algebra. For all , the following are valid:(1),(2) ,(3) if and only if ,(4) ,(5) implies , ,(6) ,(7) ,(8) ,(9) ,(10) ,(11) implies ,(12) ,,,(13) , or equivalently, ,(14) ,(15) , ,(16) , ; that is, and are involutions,(17) , ,(18) ,(19) ,(20) ,
where .

The set of all natural numbers is denoted by . We denote A -algebra is a Gödel algebra if for any . An element is involutory, if .

In this paper we will often use the identity for any (see [18]).

Definition 3 (see [2]). A nonempty subset of -algebra is called an ideal of if it satisfies:(I1),(I2) and implies for all .
A proper ideal of -algebra is called a prime ideal of if implies or for all .

Lemma 4 (see [18]). Let be an ideal in and . Then there is a prime ideal of such that and .

Definition 5 (see [18]). Let be an ideal of -algebra . is called a Gödel ideal if it satisfies for any .
A fuzzy set in is a mapping . Let be a fuzzy set in and , the set is called a level subset of .
The notations and represent two special fuzzy sets in satisfying for any and for any , respectively.
For any fuzzy sets , , () in where is an index set, we define , , and as follows: for all ,
By we mean that for all .

Definition 6 (see [15]). Let be a -algebra. A fuzzy set in is called a fuzzy ideal of if, for all ,(FI1),(FI2).

Proposition 7 (see [15]). Let be a fuzzy set in . Then is a fuzzy ideal if and only if, for each , is either empty or an ideal of .

3. Fuzzy Ideal Generated by a Fuzzy Set

In this section, we give a procedure to construct the fuzzy ideal generated by a fuzzy set. First of all we give some further properties of fuzzy ideals in -algebras. The set of all fuzzy sets in -algebra is denoted by and the set of all fuzzy ideals in is denoted by .

Proposition 8. If and , then for any . In particular, where for any .

Proof. Since , we have .

As an immediate consequence of the proposition we have the following.

Corollary 9 (see [15]). If and then for any .

Proposition 10. Let . Then for any , if and only if .

Proof. () Suppose that . By , we have Hence .
() Assume that . Since for any , , by Corollary 9, , thus .

Theorem 11. A fuzzy set in is a fuzzy ideal if and only if for all , implies .

Proof. Assuming that is a fuzzy ideal of and , then , and by Proposition 8 we have , so .
Conversely, suppose that implies for all . Since , so ; (FI1) holds. By , then , (FI2) holds.

By induction and Theorem 11 we have the following.

Corollary 12. Let be a fuzzy set in . is a fuzzy ideal if and only if, for any , implies .

Proposition 13. Letting be a -algebra and , if , , , then

Proof. From , it follows that , . Since the operation “” is isotone, so we have . While implies , hence that is, The proof is complete.

Definition 14. Let be a fuzzy set in . A fuzzy ideal is called to be generated by if and implies for any fuzzy ideal in . The fuzzy ideal generated by will be denoted by .
It is worth noticing that this definition is well-defined because is a fuzzy ideal in ; for any we have and the intersection of any family of fuzzy ideals in is a fuzzy ideal in .

Example 15 (see [7]). Let . Define ,, , and as follows: Then is a -algebra. Define a fuzzy set in by , , . It is easy to check that , .

Theorem 16. Let and be fuzzy sets in .(i)If is a fuzzy ideal in then .(ii)If then .(iii).

Proof. Trivial.

Theorem 17. Let be a fuzzy set in . If a fuzzy set in is defined as follows, for any , then .

Proof. First, we prove is a fuzzy ideal in . Suppose that (i.e., ) for any . Given any arbitrary small , there exists , such that By Proposition 13 we get It follows that Hence , and is a fuzzy ideal in by Theorem 11.
Next, since for any , it follows that , so .
Finally, supposing that is any fuzzy ideal in with , then for any ,
by Corollary 12, so .
From the above we prove .

Notation 1. In the sequel we need the notion of fuzzy points. Let be a fuzzy set in as follows: where and , and then is called a fuzzy point in with value at .

Proposition 18 (see [18]). Let be a -algebra. For any and , if , then there exists such that .

Theorem 19. Let be a fuzzy ideal in . If satisfies , , where , then

Proof. It is obvious that , so we just need to prove the converse inequality. Observe for all , Suppose we are given any fixed and an arbitrary small , it is sufficient to consider the following three cases.
Case I. There are such that(i),(ii).
Since , we obtain
Hence
Case II. There are such that(i),(ii).
By the way similar to Case I we also obtain
Case III. There are and such that(i),(ii).
Also there are and such that(iii),(iv) .
Because , by Proposition 13 and (i) we get that is,(i′).
By the similar argument and (iii) we can get(iii′).
By (i′), (iii′), and Proposition 18 there is a such that Thus so we have
This proves that . Thus .

4. Fuzzy Prime Ideals and Fuzzy Irreducible Ideals

In this section we introduce the notions of fuzzy prime ideals and fuzzy irreducible ideals and investigate their properties. The emphasis is relation between fuzzy prime ideals and fuzzy irreducible ideals.

Definition 20. A nonconstant fuzzy ideal in is called a fuzzy prime ideal in if for any , .

Theorem 21. A nonconstant fuzzy set in is a fuzzy prime ideal in if and only if where ; is a prime ideal of where ; where .

Proof. It is easy and omitted.

Example 22. Let and . If is a prime ideal of and is an proper ideal of with , then the function is a fuzzy prime ideal in where As a special case of the above example we have the following.

Example 23. If is a prime ideal of , then the characteristic function of is a fuzzy prime ideal in where

Theorem 24. Letting be a fuzzy ideal in , then is a fuzzy prime ideal in if and only if is a prime ideal of .

Proof. The “only if” part is easy. We now prove the part “if” as follows. Suppose that is a prime ideal of . where ; where , so is a prime ideal of ; where . Thus is a fuzzy prime ideal in by Theorem 21.

Theorem 25. A nonconstant fuzzy ideal in is a fuzzy prime ideal in if and only if implies or .

Proof. Suppose that is a fuzzy prime ideal in ; then is a prime ideal of by Theorem 24. If , then , and so or . Hence or .
Conversely, suppose, for any , implies or . That is, implies or , and thus is a prime ideal of . Therefore is a fuzzy prime ideal in by Theorem 24.

Note. The above theorem shows that the definition on fuzzy prime ideals in this paper and one in [15] are equivalent.

The following corollary is easy and the proof is omitted.

Corollary 26 (see [15]). A nonconstant fuzzy ideal of is a fuzzy prime ideal if and only if or for any .

We will call the next theorem as the extension theorem of fuzzy prime ideals.

Theorem 27. Let be a fuzzy prime ideal in , be a nonconstant fuzzy ideal in . If and , then is a fuzzy prime ideal.

Proof. Supposing that is a fuzzy prime ideal in , then or for any by Corollary 26. If , by and , we have , so . Likewise, if , then , so is a fuzzy prime ideal in .

In what follows we introduce another notion—fuzzy irreducible ideals, and discuss relation between fuzzy prime ideals and fuzzy irreducible ideals.

Definition 28. A nonconstant fuzzy ideal in is called a fuzzy irreducible ideal if, for any fuzzy ideals and in , implies or .

Theorem 29. Let be a nonconstant fuzzy ideal in . If is a fuzzy irreducible ideal in then is a fuzzy prime ideal in .

Proof. Suppose is a fuzzy irreducible ideal in . If is not a fuzzy prime ideal in , then there are such that . Denote and . Since , by Theorem 19 we have , but , a contradiction.

But the converse of the above theorem is not true.

Example 30 (see [18]). Let . Define , , , and as follows: Then is a -algebra. It is easy to check that , are prime ideals of .
Define a fuzzy set in by , , and then is a fuzzy prime ideal in . Indeed, where ; where ; where . Let be a fuzzy ideal in defined by , , . Let be a fuzzy ideal in defined by , . It is easy to verify that but , . Therefore is not a fuzzy irreducible ideal in .

This example shows the converse of Theorem 29 is not true.

Letting be a fuzzy set in defined by , , it is easy to check that is a fuzzy irreducible ideal in .

As is well-known, in ideal theory of -algebras, an ideal is prime if and only if it is irreducible [18]. But in the above we obtain an important fact: in fuzzy ideal theory, any fuzzy irreducible ideal is a fuzzy prime ideal, but conversely a fuzzy prime ideal may not be a fuzzy irreducible ideal.

Now we give general results.

Lemma 31. If is a fuzzy irreducible ideal in , then .

Proof. Suppose . We just need to discuss the following two cases.
Case I. where . Suppose . It is clear that . Define fuzzy sets and as follows: Obviously, and are fuzzy ideals in , and , but , , a contradiction.
Case II. where . Then there is such that , , and . Define fuzzy sets and as follows: Obviously, and are fuzzy ideals in , and , but , , a contradiction.

Lemma 32. Letting be a fuzzy prime ideal in and , then is not a fuzzy irreducible ideal in .

Proof. If , then it follows from Lemma 31 that is not a fuzzy irreducible ideal in .
If , then there are such that . Define fuzzy sets and as follows: for all Obviously, and are fuzzy ideals in , and , but , . Hence is not a fuzzy irreducible ideal in .

Lemma 33. Letting be a prime ideal of , then the characteristic function is a fuzzy irreducible ideal in .

Proof. Suppose is not a fuzzy irreducible ideal in . Then there are fuzzy ideals , in such that but , . Thus for some such that . Since and , it follows that and so ; that is, which contradicts being a prime ideal of . Therefore is a fuzzy irreducible ideal in .

In this Lemma is also a fuzzy prime ideal in . Hence this shows that under some special conditions, a fuzzy prime ideal in may be a fuzzy irreducible ideal in .

Theorem 34. Let be a fuzzy prime ideal in . Then is a fuzzy irreducible ideal in if and only if and .

Proof. () Suppose that is a fuzzy irreducible ideal in . By Lemma 31, . If , then is not a fuzzy irreducible ideal in by Lemma 32, a contradiction. Hence .
() Suppose that and . We can prove that is a fuzzy irreducible ideal in by the argument in Lemma 33.

Theorem 35. Let be a nonconstant fuzzy ideal in and let be a fuzzy point in with . Then there is a fuzzy prime ideal in satisfying and .

Proof. Since , we have . Denote Then is an ideal of and , or . We consider the following three cases.
Case I. If and , then by Lemma 4 there exists a prime ideal of such that and . Define a fuzzy set in as follows: It is easy to see that is a fuzzy irreducible ideal in with and .
Case II. If and , then the ideal does not contain . By Lemma 4 there is a prime ideal of such that and . Define a fuzzy set such that It is easy to check that is a fuzzy irreducible ideal in with and .
Case III. Suppose . We take any prime ideal of , and then . Define a fuzzy set such that It is easy to check that is a fuzzy prime ideal in with and .

Corollary 36. Let be a nonconstant fuzzy ideal of . Then is the intersection of all fuzzy prime ideals in containing .

Proof. It is immediate by Theorem 35.

This is the Krull-Stone representation theorem of fuzzy ideals in a -algebra.

5. Distributivity of Fuzzy Ideal Lattices

Before discussing the structure of fuzzy ideal lattices we first observe that for any , where , is an index set (may be infinite), , but in general, .

Example 37. Let be the -algebra defined in Example 30. It can check that , are ideals of . Define two fuzzy set as follows: where . we get the fuzzy set :
It is easy to see that , are fuzzy ideals in , but is not a fuzzy ideal in since is not an ideal of .
Now we give the following definition: for any , denote ; in general, for any ,

Theorem 38. Letting be a -algebra, then is a complete distributive lattice where and are the least lower bound and the largest upper bound of, respectively, and satisfies the following infinitely distributive law:(DL), for all .

Proof. It is easy to verify that for any ,   and are the supremum and the infimum in of and , and , so is a bounded lattice. This lattice is obviously complete.
In order to check (DL) it suffices to prove, for any , By Theorem 17, for any given and arbitrary small , there exist such that By the definition of , there are such that and thus we have so Denote and then we get Therefore Obviously we have
Thus by Proposition 8 we obtain the following.
(i) .
Since so . By a similar way we may prove that . Therefore
By Proposition 8 it follows that
(ii) .
By (i) and (ii) we have
and so
It is obvious and hence Observe that and is a fuzzy ideal of , and by Corollary 12 we get Therefore Since is arbitrary small, we have
The proof is completed.

6. Fuzzy Gödel Ideals

In this section, we introduce the notion of fuzzy Gödel ideals of -algebra and investigate some of their properties.

Definition 39. Let be a fuzzy ideal of . is called a fuzzy Gödel ideal if for all .

It is obvious that each fuzzy ideal in Gödel algebra is a fuzzy Gödel ideal. In Example 48, we will show that there exists fuzzy Gödel ideal in non-Gödel algebra.

Theorem 40. Let be a fuzzy subset of . is a fuzzy Gödel ideal if and only if, for each , is a Gödel ideal of where .

Next theorem is called the extension theorem of fuzzy Gödel ideals.

Theorem 41. Let and be fuzzy ideal of with and . If is a fuzzy Gödel ideal, then so is .

Notation 2. Let be a fuzzy ideal of . Define a fuzzy set in by

Obviously, is also a fuzzy ideal in .

Theorem 42. A fuzzy subset of is a fuzzy Gödel ideal if and only if is a fuzzy Gödel ideal of .

Proof. If is a fuzzy Gödel ideal of , then by Theorem 40, for any , is a Gödel ideal of . In particular, is a Gödel ideal of . We notice for any This shows that, for any is a Gödel ideal of where . By Theorem 40   is a fuzzy Gödel ideal of .
Conversely, suppose is a fuzzy Gödel ideal of . It is clear that and . By Theorem 41, is a fuzzy Gödel ideal of .

Theorem 43. Let be a fuzzy ideal of . The following conditions are equivalent:(i) is a fuzzy Gödel ideal,(ii) implies ,(iii) implies .

Proof. (i)(ii) Suppose that is a fuzzy Gödel ideal and . Then we have Hence .
(ii)(iii) Suppose that . Since , then we have and so . Hence and . From (ii) we get that and hence (iii)(i) Since , by (iii) we have and hence and thus is a fuzzy Gödel ideal.

Theorem 44. Let be a fuzzy ideal. is a fuzzy Gödel ideal if and only if and imply .

Proof. Suppose that is a fuzzy Gödel ideal. Let By Theorem 43(iii) we have Then it follows that and hence .
Conversely, since and , then by the assumption we have , so is a fuzzy Gödel ideal.

Theorem 45. Let be a fuzzy ideal. Then is a fuzzy Gödel ideal if and only if the following condition holds:() and imply for any .

Proof. Suppose that is a fuzzy Gödel ideal of . Letting and , then and hence . Since is a fuzzy Gödel ideal, by Theorem 43(ii) we have . () holds.
Conversely, suppose that () is true. Let and . Since then we have Since , we get that By and we get . Thus is a fuzzy Gödel ideal by Theorem 44.

Zhang et al. [15]. introduced the notion of fuzzy Boolean ideals in -algebra and proved that fuzzy Boolean ideals are equivalent to fuzzy implicative ideals. In the following, we investigate the relation between fuzzy Boolean ideals and fuzzy Gödel ideals.

Definition 46 (see [15]). A fuzzy ideal in is called a fuzzy Boolean ideal if for all .

Theorem 47. Each fuzzy Boolean ideal in is a fuzzy Gödel ideal in .

Proof. Suppose that is a fuzzy Boolean ideal in and . Then and so . By Theorem 43(ii) it follows that is a fuzzy Gödel ideal in .

But the converse of the above theorem is not true.

Example 48 (see [15]). Let . Define , , , and as follows: Then is a -algebra. Define a fuzzy set in by , ; one can easily check that is a fuzzy Gödel ideal in , but it is not a fuzzy Boolean ideal, since .

7. Conclusion

Study of fuzzy ideal theory in -algebras is technically more difficult, so far little research literature. Zhang et al. [15] initiated research in this area. In this paper we investigate further important properties of fuzzy ideals in -algebras. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals are introduced and studied. We give a procedure to generate a fuzzy ideal by a fuzzy set. Using this result we prove that any fuzzy irreducible ideal is a fuzzy prime ideal and meanwhile we give an example to show that a fuzzy prime ideal may not be a fuzzy irreducible ideal; we also give the Krull-Stone representation theorem of fuzzy ideals in -algebras. Furthermore we prove that the set of all fuzzy ideals forms a complete distributive lattice. In addition, we prove that any fuzzy Boolean ideal in -algebras is a fuzzy Gödel ideal, but the converse is not true.

In our opinion, the future study of fuzzy ideals in -algebras should be related to (1) several special types of fuzzy ideals; (2) decomposition properties of fuzzy ideals. Our obtained results can be applied in information science, engineering, computer science, and medical diagnosis.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referees for useful suggestions and comments which led to an improvement of the paper.