On Fuzzy Ideals of BL-Algebras

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 ω(0) = 1 and |Im⁡(ω)| = 2. 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.


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 [0, 1] 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 studyingalgebras. 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][6][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][14][15][16] introduced some kinds of generalized fuzzy filters in -algebras and described their relations with ordinary fuzzy filters. Another important notion ofalgebras 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 2 The Scientific World Journal 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 aalgebra 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 (0) = 1 and | Im( )| = 2. 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.

Preliminaries
Let us recall some definitions and results on -algebras.
Throughout this paper, let denote a -algebra.
Definition 3 (see [2]). A nonempty subset of -algebra is called an ideal of if it satisfies: 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 ∉ .

Corollary 12.
Let be a fuzzy set in . is a fuzzy ideal if and only if, for any , 1 , . . . ,

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 1 is a fuzzy ideal in ; for any ∈ FI( ) we have ≤ 1 and the intersection of any family of fuzzy ideals in is a fuzzy ideal in .  Proof. Trivial.
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 ∈ [0, 1], and then is called a fuzzy point in with value at .
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 > 0, it is sufficient to consider the following three cases.

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.

Theorem 21. A nonconstant fuzzy set in is a fuzzy prime ideal in if and only if
Proof. It is easy and omitted.
Example 22. Let , ∈ [0, 1] and < . If is a prime ideal of and is an proper ideal of with ⊂ , then the function : → [0, 1] 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 A, then is a fuzzy prime ideal in if and only if (0) is a prime ideal of .
Proof. The "only if " part is easy. We now prove the part "if " as Thus is a fuzzy prime ideal in by Theorem 21.
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]  We will call the next theorem as the extension theorem of fuzzy prime ideals.  In what follows we introduce another notion-fuzzy irreducible ideals, and discuss relation between fuzzy prime ideals and fuzzy irreducible ideals. This example shows the converse of Theorem 29 is not true. Letting be a fuzzy set in defined by (0) = ( ) = 1, ( ) = (1) = (0 < < 1), 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.
Proof. Suppose (0) < 1. We just need to discuss the following two cases.

Lemma 32. Letting be a fuzzy prime ideal in and | Im( )| ≥ 3, then is not a fuzzy irreducible ideal in .
Proof. If (0) ̸ = 1, then it follows from Lemma 31 that is not a fuzzy irreducible ideal in .
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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 ( ) > 0, ]( ) > 0. and so ( ∧ ) = 1; 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 .
(⇐) Suppose that (0) = 1 and | Im( )| = 2. We can prove that is a fuzzy irreducible ideal in by the argument in Lemma 33. Proof. Since ∉ , we have ( ) < . Denote Then is an ideal of and ∉ , or = 0. We consider the following three cases.
Case I. If ̸ = 0 and ̸ = 0, 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 ̸ = 0 and = 0, then the ideal {0} does not contain . By Lemma 4 there is a prime ideal of such that {0} ⊆ and ∉ . Define a fuzzy set ] such that It is easy to check that ] is a fuzzy irreducible ideal in with ≤ ] and ∉ ].
Case III. Suppose = 0. We take any prime ideal of , and then 0 ∈ . Define a fuzzy set ] such that It is easy to check that ] is a fuzzy prime ideal in with ≤ ] and 0 ∉ ].

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.

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 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.
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 ∈ [0, 1], is a Gödel ideal of . In particular, (0) = { ∈ | ( ) = (0)} is a Gödel ideal of . We notice for any ∈ [0, 1] This shows that, for any ∈ [0, 1], ( ) is a Gödel ideal of where ( ) ̸ = 0. By Theorem 40 is a fuzzy Gödel ideal of .
Conversely, suppose is a fuzzy Gödel ideal of . It is clear that ≤ and (0) = (0). 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, Hence and so (( and and hence 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
Proof. Suppose that is a fuzzy Gödel ideal. Let By Theorem 43(iii) we have Then it follows that and hence ((( − ) 2 → − ) − ) = (0). Since is a fuzzy Gödel ideal, by Theorem 43(ii) we have ( then we have Since (( − → ( − → − )) − ) = (0), we get that 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.
Theorem 47. Each fuzzy Boolean ideal in is a fuzzy Gödel ideal in .
But the converse of the above theorem is not true.

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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 inalgebras 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.