# Fuzzy Prime Ideal Theorem in Residuated Lattices

Authors: Pierre Carole Kengne; Blaise Blériot Koguep; Celestin Lele
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

This 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.


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## Body

## 1. 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 [4–8]. The 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. The notion of fuzzy ideal has been studied in several structures such as rings [12], lattices [13, 14], MV-algebras [15], BL-algebras [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.The remainder of this paper is organized as follows: Section2 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.


## 2. Review on Residuated Lattices and Ideals

Definition 1.
(see [3, 17]). A residuated lattice is an algebraic structure A=A,∧,∨,⊙,⟶,0,1 of type 2,2,2,2,0,0 satisfying the following axioms:(R1)A,∧,∨,0,1 is a bounded lattice(R2)A,⊙,1 is a commutative monoid(R3) For alla,b,c∈A, c≤a⟶b if and only if a⊙c≤b
Let us give the following notations in a residuated latticeA:(i)
∀a∈A, a′:=a⟶0 and a′′=a″(ii)
∀a,b∈A, a⊘b:=a′⟶bDefinition 2.
(see [1, 18]). A residuated lattice A=A,∧,∨,⊙,⟶,0,1 is an MTL-algebra if it satisfies the prelinearity condition, that is, a⟶b∨b⟶a=1, for all a,b∈A.Definition 3.
(see [19]). A De Morgan residuated lattice is a residuated lattice A=A,∧,∨,⊙,⟶,0,1 such that, for all a,b∈A, a∧b′=a′∨b′.Example 1.
LetA be a lattice defined by the Hass diagram of Figure 1.
Define⟶ and ⊙ as follows:
A,∧,∨,⊙,⟶,0,1 is an MTL-algebra.Figure 1
Hass diagram.Example 2.
LetA be a lattice defined by the Hass diagram of Figure 2.
Define⟶ and ⊙ as follows:
A,∧,∨,⊙,⟶,0,1 is not a De Morgan residuated lattice. We have a∧d′=0′=1 and a′∨d′=d∨a=m; then a∧d′≠a′∨d′.
Any MTL-algebra is a De Morgan residuated lattice but the converse is not always true.Figure 2
Example of lattice 2.Example 3.
LetA be a lattice defined by the Hass diagram of Figure 3.
Define⟶ and ⊙ as follows:
A,≤,⊙,⟶,0,1 is a De Morgan residuated lattice. Since a⟶b∨b⟶a=f∨e=m≠1, it follows that A is not an MTL-algebra.Figure 3
Hass diagram.Theorem 1 (see [8]).
For any residuated latticeA=A,∧,∨,⊙,⟶,0,1, the following properties hold for every x,y,z∈A:(P1)x⊙y⟶z=x⟶y⟶z(P2) Ifx≤y, then y′≤x′(P3)y⟶z≤x⟶y⟶x⟶z(P4)x⊙y′=x⟶y′(P5)1⟶x=x, x⟶x=1(P6)x⟶y⟶z=y⟶x⟶z;(P7)0′=1,1′=0,x′=x‴,x≤x″(P8)y⟶x≤x⟶z⟶y⟶z(P9)x⟶y∧z=x⟶y∧x⟶z(P10)x∨y⟶z=x⟶z∧y⟶zDefinition 4 (see [6]).
LetA=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:(F1) For everya,b∈F, a∧b∈F(F2) For everya,b∈A, if a≤b and a∈F, b∈FDefinition 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:(I1) For everya,b∈I, a⊘b∈I(I2) For everya,b∈A, if a≤b and b∈I, then a∈I
We denote byℐA the set of all ideals of A. An ideal I is called proper if I≠A. If I∈ℐA, then 0∈I and x∈I if x″∈I.
There are two types of prime ideal in any residuated lattice.Definition 6.
(see [10, 20]). Let I be a proper ideal of a residuated lattice A. I is said to be a prime ideal, if, for all I1,I2∈ℐA, if I1∩I2⊆I, then I1⊆I and I2⊆I.Proposition 1 (see [10, 20]).
LetP be a subset of residuated lattice A. P is a prime ideal if and only if, for all a,b∈A, if a″∧b″∈P, then a∈P or b∈P.Remark 1.
(see [19]). If A is a De Morgan residuated lattice, then I∈ℐA is a prime ideal if, for all a,b∈A, a∧b∈I implies a∈I or b∈I.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.
The next theorem gives the relation between these two types of prime ideal.Theorem 2 (see [8]).
LetA 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]).
LetI 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]).
LetA 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=∅.


## 3. Fuzzy Ideal Generated by a Fuzzy Subset of Residuated Lattice

LetA=A,∧,∨,⊙,⟶,0,1 be a residuated lattice.We first recall some definitions and properties of fuzzy subset.Definition 9.
(see [11]). Let A be a nonempty set.(i)
A mapη:A⟶0,1 is called a fuzzy subset of A(ii)
A fuzzy subsetη of A is called proper if it is not a constant map(iii)
Ifη:A⟶0,1 is a fuzzy subset of A and t∈0,1, then ηt=a∈A/ηa≥t is called the t-cut set of ηDefinition 10.
(see [8]). Let η be a fuzzy subset of A. η is a fuzzy ideal of A, if it satisfies the following conditions:(FI1) for anya,b∈A, if a≤b, then ηa≥ηb(FI2) for anya,b∈A, ηa⊘b≥minηa,ηbExample 4.
LetA be a lattice defined by the Hass diagram of Figure 4.
Define⟶ and ⊙ as follows (Table 1):
A=A,≤,⊙,⟶,0,1 is a residuated lattice. The fuzzy subset μ of A, defined by(1)μx=12,ifx=0,13,ifx=b,15,ifx∈1,c,a,is a fuzzy ideal of A.Figure 4
Example lattice 2.Table 1
Table of operations⊙ and ⟶ defined on A.Definition 11.
(see [6, 13]). Let η be a fuzzy subset of A.(1)
η is a fuzzy filter of latticeA,∧,∨,0,1, if it satisfies the following conditions:(i)
for anya,b∈A, if a≤b, then ηa≤ηb(ii)
for anya,b∈A, ηa∧b=minηa,ηb(2)
η is a fuzzy filter of residuated latticeA, if it satisfies the following conditions:(i)
for anya,b∈A, if a≤b, then ηa≤ηb(ii)
for anya,b∈A, ηa⊙b=minηa,ηb
Note that any fuzzy filter of residuated latticeA is a fuzzy filter of lattice A,∧,∨,0,1; but the converse is not always true.Example 5.
Let us consider the residuated latticeA defined in Example 3. The fuzzy subset θ of A, defined by(2)θx=0,ifx=0,14,ifx∈a,n,12,ifx∈b,d,34,ifx∈c,m,1,ifx=1,is a fuzzy filter of lattice A,≤ but it is not a filter of a residuated lattice A because θn⊙n=θ0≠θn.
Letη,θ be two fuzzy subsets of A. We define the order relation ≤ by η≤θ⟺ηa≤θa, for all a∈A and, for any family ηi;i∈Λ of fuzzy ideals (or fuzzy filters) of A, ∨i∈Ληia=supi∈Ληia and ∧i∈Ληia=infi∈Ληia.
LetFIA and FFA denote, respectively, the set of fuzzy ideals of residuated lattice A and the set of fuzzy filters of lattice A.Proposition 3.
Letηi;i∈Λ be a family of fuzzy ideals of a residuated lattice A. Then μ=∧i∈Ληi is a fuzzy ideal of A.Proof.
Leta,b∈A such that a≤b. We have ηia≥ηib; then μa≥μb.
Moreover, for alla,b∈A, we have ηia⊘b≥minηia,ηib; then μa⊘b=infi∈Ληia⊘b≥infi∈Λminηia,ηib=minμa,μb. Thus, μ is a fuzzy ideal of A.
In general,∨i∈Ληi 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,(3)η1x=1,ifx=0,910,ifx=b,310,ifx∈1,c,a,η2x=1,ifx=045,ifx=a,35,ifx∈1,c,b,of residuated lattice A defined in Example 4. For η=η1∨η2, the fuzzy subset of A, we have(4)ηx=1,ifx=0,45,ifx=a,910,ifx=b,35,ifx∈1,c,which is not a fuzzy ideal of A because ηa⊘b=η1=3/5≱ηa∧ηb=4/5.
Letα∈0;1; we define Cα:A⟶0;1 by Cαa=α, for all a∈A.Proposition 4 (see [7]).
Letη be a fuzzy ideal of A and α∈0;1 such that α<η0. Then the fuzzy subset η∨Cα of A is a fuzzy ideal of A.Theorem 4 (see [8]).
Letη be a fuzzy subset of A. η is a fuzzy ideal of A if and only if, for each t∈L, ηt≠∅ implies that ηt is an ideal of A.Notation 1.
In the remainder of this paper, the following map will be very useful. LetI be a nonempty subset of A and α,β∈0;1 such that α>β. Define the map ηIα,β as follows: ηIα,βa=α,ifa∈Iβ,otherwise.Proposition 5 (see [8]).
ηIα,β is a proper fuzzy ideal of A if and only if I is a proper ideal of A.Definition 12.
Letη: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. The fuzzy ideal generated by η will be denoted by η^.
Letη be a fuzzy subset of A. For all a∈A,a∈ηηa⊆η^ηa and ηa∈0;1; then t∈0;1/a∈ηt^≠∅.Theorem 5.
Letη be a fuzzy subset of A. Then the fuzzy subset η∗ of A defined by η∗a=supt∈0;1/a∈ηt^, for all a∈A, is the fuzzy ideal generated by η.Proof.
Letα∈0;1. Suppose that α∈Imη∗; take αn=α−1/n, for all n∈ℕ∗ and a∈ηα∗. We have η∗a≥α≥αn; then there exists t0∈0;1 such that a∈η^t0 and t0≥αn, for all n∈ℕ∗. That is, a∈η^t0 and ηt0⊆ηαn, for all n∈ℕ∗. Then a∈η^t0⊆η^αn, for all n∈ℕ∗, which implies that a∈∩n∈ℕ∗η^αn. Thus, ηα∗⊆∩n∈ℕ∗η^αn. If a∈∩n∈ℕ∗η^αn, then, for all n∈ℕ∗, αn∈t∈0;1/a∈ηt^; therefore, for all n∈ℕ∗, η∗a≥αn=α−1/n, which implies that η∗a≥α; that is, x∈ηα∗. Thus, ηα∗=∩n∈ℕ∗η^αn. By Proposition 3, ηα∗ is an ideal of A. If α∉Imη∗, then ηα∗=ηβ∗, where β=supt∈Imη∗/t>α∈Imη∗ or ηα∗=∅. Therefore, by Theorem 4, η∗ is a fuzzy ideal of A.
Leta∈A; we have a∈ηα, where α=ηa; thus, a∈η^α; that is, α∈t∈0;1/a∈η^t. Therefore, ηa=α≤η∗a; that is, η≤η∗.
Letθ be a fuzzy ideal of A such that η≤θ and a∈A. If η∗a=0, then η∗a≤θa. Suppose that η∗a=α≠0. Then a∈ηα∗=∩n∈ℕ∗η^αn; that is, for all n∈ℕ∗, ηa≥αn=α−1/n. Since θa≥ηa≥α=η∗a, θ≥η∗. Hence, η∗=η^.Example 7.
Let us consider the residuated latticeA defined in Example 4. Let us consider the fuzzy subset η of A defined by(5)ηx=12,ifx=0,13,ifx=b,15,ifx=a,110,ifx∈1,c.
Lett∈0;1; if t∈0;1/10, then ηt=A=ηt^. If t∈1/10;1/5, then ηt=0,a,b and ηt^=A. If t∈1/5;1/3, then ηt=0,b=ηt^. If t∈1/3;1/2, then ηt=0=ηt^. If t∈1/2;1, then ηt=∅. Therefore, we haveη∗0=supt∈0;1/0∈ηt^=1/2η∗a=supt∈0;1/a∈ηt^=1/5η∗b=supt∈0;1/b∈ηt^=1/3η∗c=supt∈0;1/c∈ηt^=1/5η∗1=supt∈0;1/1∈ηt^=1/5
Thus,(6)η∗x=12,ifx=0,13,ifx=b,15,ifx∈1,c,a,


## 4. Fuzzy Prime Ideal Theorem

Definition 13.
A proper fuzzy idealη of a residuated lattice A is said to be fuzzy prime if ηx″∧y″=ηx or ηx″∧y″=ηy, for any x,y∈A.Proposition 6.
Letη be a proper fuzzy subset of a residuated lattice A. η is a fuzzy prime ideal of A if and only if, for any t∈0;1, if ∅≠ηt≠A, then ηt is a prime ideal of A.Proof.
Letη be a proper fuzzy subset of A. Suppose that η is a fuzzy prime ideal. Let t∈0;1 such that ∅≠ηt≠A. By Theorem 4, ηt is an ideal of A. Let x,y∈A such that x″∧y″∈ηt; that is, ηx″∧y″≥t; then ηx≥t or ηy≥t; that is, x∈ηt or y∈ηt. Therefore, ηt is a prime ideal of A.
Conversely, suppose that, for allt∈0;1, if ηt is nontrivial, then ηt is a prime ideal. By Theorem 4, η is a fuzzy ideal of A. Let x,y∈A. Take α=ηx″∧y″; we have ηx=ηx′′≠ηx″∧y″ or ηy=ηy″≠ηx″∧y″ because if it is not, then ηx=ηy, which is absurd (η is proper); that is, ηx<ηx″∧y″=α or ηy<ηx″∧y″=α; that is, x∉ηα or y∉ηα. Thus, ηα≠A. Moreover, ηα≠∅ because x″∧y″∈ηα; then by hypothesis ηα is a prime ideal and x∈ηα or y∈ηα; that is, ηx≥ηx″∧y″ or ηy≥ηx″∧y″. In addition, ηx″∧y″≥ηx″=ηx and ηx″∧y″≥ηy″=ηy because η is a fuzzy ideal. Therefore, ηx″∧y″=ηx or ηx″∧y″=ηy.
IfA is a De Morgan residuated lattice, then η is a fuzzy prime ideal of A if ηx∧y=ηx or ηx∧y=ηy, for any x,y∈A.Definition 14.
A fuzzy idealη of a residuated lattice A is said to be fuzzy prime of the second kind if it is nonconstant and ηx⟶y′=η0 or ηy⟶x′=η0 for any x,y∈A.Example 8.
LetA be the residuated lattice defined in Example 4. Then, η, defined by(7)ηx=15,ifx∈0,a,110,ifx∈1,c,b,is a fuzzy prime ideal of the second kind of A.Lemma 1.
LetA be a residuated lattice. For all x,y∈A, we have x″∧y″⊘x⟶y′≥x and x″∧y″⊘y⟶x′≥y.Proof.
Letx,y∈A. We have x″∧y″⊘x⟶y′=x⟶y⟶x″∧y″″ by property (P4) of Theorem 1. Since x″∧y″″≥x″∧y″≥x∧y, then x″∧y″⊘x⟶y′≥x⟶y⟶x∧y=x⟶y⟶x∧x⟶y⟶y≥x∧x=x. Thus, x″∧y″⊘x⟶y′≥x. In the same way, we show that x″∧y″⊘y⟶x′≥y.Theorem 6.
Any fuzzy prime ideal of the second kind of residuated latticeA is a fuzzy prime ideal of A. If A is an MTL-algebra, then the notions of fuzzy prime ideal of A and fuzzy prime ideal of the second kind of A are equivalent.Proof.
Letη be a fuzzy prime ideal of the second kind of A. Let x,y∈A; we have ηx⟶y′=η0 or ηy⟶x′=η0. By Lemma 1, x″∧y″⊘x⟶y′≥x and x″∧y″⊘y⟶x′≥y; then ηx≥ηx″∧y″⊘x⟶y′≥minηx″∧y″;ηx⟶y′ and ηy≥ηx″∧y″⊘y⟶x′≥minηx″∧y″;ηy⟶x′. If ηx⟶y′=η0, then ηx≥minηx″∧y″;η0=ηx″∧y″. From the fact that η is a fuzzy ideal, we have ηx″∧y″≥ηx; then ηx=ηx″∧y″. Identically, if ηy⟶x′=η0, then ηy=ηx″∧y″. In conclusion, η is a fuzzy prime ideal of A.
Assume thatA is an MTL-algebra and let η be a fuzzy prime ideal of A. Then, A satisfies the prelinearity condition; that is, for any x,y∈A, we have x⟶y∨y⟶x=1; that is, x⟶y′∧y⟶x′=0; then η0=ηx⟶y″′∧y⟶x″′. Therefore, η0=ηx⟶y′ or η0=ηy⟶x′, by hypothesis. Thus, 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. The proof of this statement is given by the following counterexample.Example 9.
Let us consider the residuated latticeA defined in Example 3 and let η be the fuzzy ideal of A defined by(8)ηx=12,ifx∈1,m,c,d,b,85,ifx∈a,n,1,ifx=0.
η is a fuzzy prime ideal of A which is not a fuzzy prime ideal of the second kind of A, because ηa⟶b′=ηa≠η0 and ηb⟶a′=ηb≠η0.Definition 15.
A proper fuzzy idealη 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 Example3.
Consider that the fuzzy subsetη, defined by(9)ηx=12,ifx∈1,m,c,d,b,1,ifx∈0,a,n,is a prime fuzzy ideal of A.Remark 2.
A fuzzy prime ideal ofA is not necessarily a prime fuzzy ideal of A as the following example shows.Example 11.
Let us consider the residuated lattice defined in Example3. Let η be a fuzzy subset of A defined by(10)ηx=12,ifx∈1,m,c,d,b,85,ifx∈0,a,n.
η is a fuzzy prime ideal of A. Let μ and θ be two fuzzy ideals of A defined by(11)μx=15,ifx∈1,m,c,d,b,1,ifx∈0,a,n,θx=35,ifx∈1,m,c,d,b,710,ifx∈0,a,n.
We have(12)μ∧θx=15,ifx∈1,m,c,d,b,710,ifx∈0,a,n.
Thenμ∧θ≤η, μ≰η, and θ≰η; therefore, η is not a prime fuzzy ideal of A.Definition 16.
Letη:A⟶0,1 be a proper fuzzy ideal of A. η is called fuzzy maximal ideal if, for all t∈0,1, ∅≠ηt≠A implies that ηt is a maximal ideal of A.Example 12.
Let us consider the residuated lattice defined in Example3 and let η be a fuzzy subset of A defined by(13)ηx=0,ifx∈1,m,c,d,b,1,ifx∈0,a,n,
η is a fuzzy maximal ideal of A.Proposition 7.
Letη 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.
Letη be a fuzzy maximal ideal of A. Then, for all t∈0,1 such that ∅≠ηt≠A, ηt is a maximal ideal of A. By Proposition 2, ηt is a prime ideal of A for all t∈0,1 such that ∅≠ηt≠A. Thus, according to Proposition 6, η is a fuzzy prime ideal of A.
The converse of the above proposition is not true. Let us consider the residuated lattice defined in Example3 and let η be a fuzzy subset of A defined by(14)ηx=15,ifx∈A,0,710,ifx=0.
η is a fuzzy prime ideal of A, which is not a fuzzy maximal ideal of A because η3/5=0 is not maximal.Lemma 2.
Lett∈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. Then I=x∈A|ηx>t and J=x∈A|μx>t are, respectively, the ideal of the residuated lattice A and the filter of the lattice A,∧,∨,0,1.Proof.
Letx∈A and y∈I such that x≤y. Then ηx≥ηy>t; that is, x∈I. Let x,y∈I. We have ηx′⊙y′′=ηx′⟶y″≥ηx∧ηy″. Since η is a fuzzy ideal of A, ηy″=ηy. Therefore, ηx′⊙y′′≥ηx∧ηy>t because x,y∈I. Thus, x′⊙y′′∈I. Hence, I is an ideal of A.
Letx∈J and y∈A such that x≤y. Then μy≥μx>t; that is, y∈J. Let x,y∈J; we have μx∧y≥μx∧μy>t. Therefore, x∧y∈J. Thus, J is a filter of the lattice A,∧,∨,0,1.Theorem 7.
Lett∈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 such that η∧μ≤t. Then there exists a fuzzy prime ideal θ of A such that η≤θ and μ∧θ≤t.Proof.
LetI=x∈A|ηx>t and J=x∈A|μx>t. 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 thatI∩J=∅. Suppose that there exists x∈I∩J. Then ηx>t and μx>t. Therefore, η∧μx>t; this is a contradiction because, by hypothesis, η∧μ≤t. Thus, I∩J=∅.
Using the fact thatI is an ideal of the residuated lattice A, J is a filter of lattice A,∧,∨,0,1 and I∩J=∅ and, by Theorem 4, there exists a prime ideal P of the residuated lattice A such that I⊆P and P∩J=∅. Let(15)θx=1,ifx∈P,t,ifx∉P.
We haveθ=ηP1,t; then, by Proposition 5, θ is a fuzzy ideal of A. If x″∧y″∈P, then x∈P or y∈P; that is, θx=1 or θy=1; therefore, θx=θx″∧y″ or θy=θx″∧y″. If x″∧y″∉P, then θx″∧y″=t≤θx,θy because θx,θy∈t,1. In addition, θ is a fuzzy ideal of A and, for all x,y∈A, x″∧y″≤x″,y″; then, θx″∧y″≥θx″,θy″. Therefore, θx″∧y″=θx or θx″∧y″=θy, since θx″=θx and θy″=θy. Thus, θ is a fuzzy prime ideal of A.
Letx∈A; if x∈P, then θx=1≥ηx. Else, x∉P; then x∉I and θx=t, because I⊆P; in this case, ηx<t=θx. Thus, η≤θ.
Letx∈A; if x∈P, then θx=1 and x∉J (because P∩J=∅). Therefore, μ∧θx=μx<t. If x∉P, then μ∧θx=μx∧t<t. Thus, μ∧θ≤t.


## 5. 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. The last result of this paper is the fuzzy prime ideal theorem.



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