Fuzzy Prime Ideal Theorem in Residuated Lattices

)is paper mainly focuses on building the fuzzy prime ideal theorem of residuated lattices. Firstly, we introduce the notion of fuzzy ideal generated by a fuzzy subset of a residuated lattice and we give a characterization. Also, we introduce different types of fuzzy prime ideals and establish existing relationships between them.We prove that any fuzzy maximal ideal is a fuzzy prime ideal in residuated lattice. Finally, we give and prove the fuzzy prime ideal theorem in residuated lattice.


Introduction
Nonclassical logic is closely related to logic algebraic systems. Many researches have motivated to develop nonclassical logic and also enrich the content of algebra [1,2]. In modern fuzzy logic theory, residuated lattices and some related algebraic systems play an extremely important role because they provide algebraic frameworks to fuzzy logic and fuzzy reasoning.
Ward and Dilworth [3] initiated the notion of residuated lattice and it interested other authors [4][5][6][7][8]. e notion of ideal has been introduced in several algebraic structures such as BL-algebras [9] and residuated lattice [5,8]. Piciu [10] gives and proves the prime ideal theorem in residuated lattices. Dealing with certain and uncertain information is an important task of the artificial intelligence, in order to make computer simulate human being. To handle such information, Zadeh [11] introduced the notion of fuzzy subset of a nonempty set A as a function μ: A ⟶ I, where I � [0; 1] is the unit interval of real numbers. Since then, a lot of works have been done on fuzzy mathematical structures and most authors used the above original definition of a fuzzy set. e notion of fuzzy ideal has been studied in several structures such as rings [12], lattices [13,14], MV-algebras [15], BLalgebras [16], and residuated lattices [6,8,17]. However, recent work of Piciu [10] gives the prime ideal theorem in residuated lattices. But that theorem is not yet investigated in fuzzy logic. In this work, we give and demonstrate the fuzzy prime ideal theorem in residuated lattice. e remainder of this paper is organized as follows: Section 2 is a review on residuated lattices and ideals, whereas Section 3 contains the characterization of a fuzzy ideal generated by a fuzzy subset in residuated lattice. In Section 4, we study the different types of fuzzy prime ideals in residuated lattice, and we give some relations between them. We also define the notion of fuzzy maximal ideal and we give the fuzzy prime ideal theorem of residuated lattice.
Define ⟶ and ⊙ as follows: Example 2. Let A be a lattice defined by the Hass diagram of Figure 2. Define ⟶ and ⊙ as follows: Theorem 1 (see [8]). For any residuated lattice A � (A, ∧, ∨, ⊙ , ⟶ , 0, 1), the following properties hold for every x, y, z ∈ A: Definition 4 (see [6]). Let A � (A, ≤ ) be a lattice and let F be a nonempty subset of A. We say that F is a filter of A, if it satisfies the following conditions: Definition 5 (see [5,8,20]). Let A � (A, ∧, ∨, ⊗ , ⟶ , 0, 1) be a residuated lattice and let I be a nonempty subset of A. We say that I is an ideal of A if it satisfies the following conditions: ere are two types of prime ideal in any residuated lattice.
Remark 1 (see [19]). If A is a De Morgan residuated lattice, Definition 7 (see [20]). Let I be a proper ideal of a residuated lattice A. I is said to be a prime ideal of second kind, if and only if, for any a, b ∈ A, (a ⟶ b) ′ ∈ I or (b ⟶ a) ′ ∈ I. e next theorem gives the relation between these two types of prime ideal.
Theorem 2 (see [8]). Let A be a residuated lattice. Every prime ideal of second kind of A is also a prime ideal. If A is an MTL-algebra, then prime ideal of A and prime ideal of second kind of A are equivalent.
Let us recall the notion of maximal ideal.
Definition 8 (see [20]). Let I be a proper ideal of a residuated lattice A. I is said to be a maximal ideal, if, for any ideal J of A, I ⊆ J implies that J � I or J � A.
Proposition 2 (see [10]). Let I be an ideal of A. If I is a maximal ideal of A, then it is a prime ideal of A.
Theorem 3 (see [10]). Let A be a residuated lattice. If I is an ideal of A and J is a filter of the lattice (A, ∧, ∨, 0, 1) such that I ∩ J � ∅, then there is a prime ideal P of A such that I ⊆ P and P ∩ J � ∅.

Fuzzy Ideal Generated by a Fuzzy Subset of Residuated Lattice
Let A � (A, ∧, ∨, ⊙ , ⟶ , 0, 1) be a residuated lattice. We first recall some definitions and properties of fuzzy subset.
Definition 10 (see [8]). Let η be a fuzzy subset of A. η is a fuzzy ideal of A, if it satisfies the following conditions: Example 4. Let A be a lattice defined by the Hass diagram of Figure 4. Define ⟶ and ⊙ as follows (Table 1): is a fuzzy ideal of A.
(1) η is a fuzzy filter of lattice (A, ∧, ∨, 0, 1), if it satisfies the following conditions:  (2) η is a fuzzy filter of residuated lattice A, if it satisfies the following conditions: Note that any fuzzy filter of residuated lattice A is a fuzzy filter of lattice (A, ∧, ∨, 0, 1); but the converse is not always true.
Example 5. Let us consider the residuated lattice A defined in Example 3. e fuzzy subset θ of A, defined by is a fuzzy filter of lattice (A, ≤ ) but it is not a filter of a residuated lattice A because θ(n ⊙ n) � θ(0) ≠ θ(n). Let η, θ be two fuzzy subsets of A. We define the order relation ≤ by η ≤ θ ⟺ η(a) ≤ θ(a), for all a ∈ A and, for any family η i ; i ∈ Λ of fuzzy ideals (or fuzzy filters) of A, Let FI(A) and FF(A) denote, respectively, the set of fuzzy ideals of residuated lattice A and the set of fuzzy filters of lattice A.
In general, ∨ i∈Λ η i of fuzzy ideals of A is not always a fuzzy ideal of A. Indeed, this following example shows it. □ Example 6. Let us consider the fuzzy ideals, Proposition 4 (see [7]). Let η be a fuzzy ideal of A and α ∈ [0; 1] such that α < η(0). en the fuzzy subset η ∨ C α of A is a fuzzy ideal of A.
Theorem 4 (see [8]). Let η be a fuzzy subset of A. η is a fuzzy ideal of A if and only if, for each t ∈ L, η t ≠ ∅ implies that η t is an ideal of A. Notation 1. In the remainder of this paper, the following map will be very useful. Let I be a nonempty subset of A and Proposition 5 (see [8])

. η I α,β is a proper fuzzy ideal of A if and only if I is a proper ideal of A.
Definition 12. Let η: A ⟶ [0; 1] be a fuzzy subset of A. A fuzzy ideal θ of A is said to be generated by η if η ≤ θ and, for any fuzzy ideal μ of A, if η ≤ μ, then θ ≤ μ. e fuzzy ideal generated by η will be denoted by η.
Theorem 5. Let η be a fuzzy subset of A. en the fuzzy subset η * of A defined by η * (a) � sup t ∈ [0; 1]/a ∈ η t , for all a ∈ A, is the fuzzy ideal generated by η.

Proposition 6. Let η be a proper fuzzy subset of a residuated lattice A. η is a fuzzy prime ideal of A if and only if, for any
Proof. Let η be a proper fuzzy subset of A. Suppose that η is a fuzzy prime ideal. Let t ∈ [0; 1] such that ∅ ≠ η t ≠ A. By eorem 4, η t is an ideal of A. Let x, y ∈ A such that x ″ ∧y ″ ∈ η t ; that is, η(x ″ ∧y ″ ) ≥ t; then η(x) ≥ t or η(y) ≥ t; that is, x ∈ η t or y ∈ η t . erefore, η t is a prime ideal of A.
In the same way, we show that (x ″ ∧y ″ )⊘(y ⟶ x) ′ ≥ y. Proof. Let η be a fuzzy prime ideal of the second kind of A.
Assume that A is an MTL-algebra and let η be a fuzzy prime ideal of A. en, A satisfies the prelinearity condition; that is, for any , by hypothesis. us, η is a fuzzy prime ideal of the second kind of A.
In general, a fuzzy prime ideal of a residuated lattice is not always a fuzzy prime ideal of the second kind, unless the residuated lattice is an MTL-algebra. e proof of this statement is given by the following counterexample. □ Example 9. Let us consider the residuated lattice A defined in Example 3 and let η be the fuzzy ideal of A defined by η is a fuzzy prime ideal of A which is not a fuzzy prime ideal of the second kind of A, because η ((a ⟶ b)

Definition 15. A proper fuzzy ideal η of a residuated lattice
A is prime fuzzy if, for fuzzy ideals μ and θ of A, θ ∧ μ ≤ η implies that θ ≤ η or μ ≤ η.
Example 10. Let us consider the residuated lattice defined in Example 3.
Consider that the fuzzy subset η, defined by is a prime fuzzy ideal of A.

Remark 2.
A fuzzy prime ideal of A is not necessarily a prime fuzzy ideal of A as the following example shows.
Example 11. Let us consider the residuated lattice defined in Example 3. Let η be a fuzzy subset of A defined by η is a fuzzy prime ideal of A. Let μ and θ be two fuzzy ideals of A defined by We have 6 International Journal of Mathematics and Mathematical Sciences en μ ∧ θ ≤ η, μ ≰ η, and θ ≰ η; therefore, η is not a prime fuzzy ideal of A.
Example 12. Let us consider the residuated lattice defined in Example 3 and let η be a fuzzy subset of A defined by η is a fuzzy maximal ideal of A.

Proposition 7. Let η be a fuzzy ideal of A. If η is a fuzzy maximal ideal of A, then η is a fuzzy prime ideal of A.
Proof. Let η be a fuzzy maximal ideal of A. en, for all t ∈ [0, 1] such that ∅ ≠ η t ≠ A, η t is a maximal ideal of A. By Proposition 2, η t is a prime ideal of A for all t ∈ [0, 1] such that ∅ ≠ η t ≠ A. us, according to Proposition 6, η is a fuzzy prime ideal of A. e converse of the above proposition is not true. Let us consider the residuated lattice defined in Example 3 and let η be a fuzzy subset of A defined by η is a fuzzy prime ideal of A, which is not a fuzzy maximal ideal of A because η 3/5 � 0 { } is not maximal. □ Lemma 2. Let t ∈ [0; 1], let η be a fuzzy ideal of a residuated lattice A, and let μ be a fuzzy filter of the lattice (A, ∧, ∨, 0, 1).
en I � x ∈ A|η(x) > t and J � x ∈ A | μ(x) > t are, respectively, the ideal of the residuated lattice A and the filter of the lattice (A, ∧, ∨, 0, 1).
Proof. Let x ∈ A and y ∈ I such that x ≤ y.
us, (x ′ ⊙ y ′ ) ′ ∈ I. Hence, I is an ideal of A.
By Lemma 2, I is an ideal of the residuated lattice A and J is a filter of the lattice (A, ∧, ∨, 0, 1).
Let us show that I ∩ J � ∅. Suppose that there exists x ∈ I ∩ J.

Conclusion
In this paper, we have investigated the notion of fuzzy ideal generated by a fuzzy subset of residuated lattice and established the fuzzy prime ideal theorem of this structure. Besides, we have studied different types of fuzzy prime ideals (fuzzy prime ideal, fuzzy prime ideal of second kind, and prime fuzzy ideal) and fuzzy maximal ideal of residuated lattice. We have proved that any fuzzy maximal ideal is a fuzzy prime ideal, but the converse is not always true. Moreover, any prime fuzzy ideal and any fuzzy prime ideal of the second kind are fuzzy prime ideals and the converse is not also true. We have also given a characterization of fuzzy ideal generated by a fuzzy subset of residuated lattice. e last result of this paper is the fuzzy prime ideal theorem.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.
International Journal of Mathematics and Mathematical Sciences 7