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.
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 as a function , where 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: 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.
2. Review on Residuated Lattices and Ideals
Definition 1. (see [3, 17]). A residuated lattice is an algebraic structure of type satisfying the following axioms: (R1) is a bounded lattice (R2) is a commutative monoid (R3) For all , if and only if Let us give the following notations in a residuated lattice :(i), and (ii),
Definition 2. (see [1, 18]). A residuated lattice is an MTL-algebra if it satisfies the prelinearity condition, that is, , for all .
Definition 3. (see [19]). A De Morgan residuated lattice is a residuated lattice such that, for all , .
Example 1. Let be a lattice defined by the Hass diagram of Figure 1.
Define and as follows:
is an MTL-algebra.
Example 2. Let be a lattice defined by the Hass diagram of Figure 2.
Define and as follows:
is not a De Morgan residuated lattice. We have and ; then .
Any MTL-algebra is a De Morgan residuated lattice but the converse is not always true.
Example 3. Let be a lattice defined by the Hass diagram of Figure 3.
Define and as follows:
is a De Morgan residuated lattice. Since , it follows that is not an MTL-algebra.
Theorem 1 (see [8]). For any residuated lattice , the following properties hold for every : (P1) (P2) If , then (P3) (P4) (P5) , (P6) ; (P7) (P8) (P9) (P10)
Definition 4 (see [6]). Let be a lattice and let be a nonempty subset of . We say that is a filter of , if it satisfies the following conditions: (F1) For every , (F2) For every , if and ,
Definition 5. (see [5, 8, 20]). Let be a residuated lattice and let be a nonempty subset of . We say that is an ideal of if it satisfies the following conditions: (I1) For every , (I2) For every , if and , then We denote by the set of all ideals of . An ideal is called proper if . If , then and if .
There are two types of prime ideal in any residuated lattice.
Definition 6. (see [10, 20]). Let be a proper ideal of a residuated lattice . is said to be a prime ideal, if, for all , if , then and .
Proposition 1 (see [10, 20]). Let be a subset of residuated lattice . is a prime ideal if and only if, for all , if , then or .
Remark 1. (see [19]). If is a De Morgan residuated lattice, then is a prime ideal if, for all , implies or .
Definition 7. (see [20]). Let be a proper ideal of a residuated lattice . is said to be a prime ideal of second kind, if and only if, for any , or .
The next theorem gives the relation between these two types of prime ideal.
Theorem 2 (see [8]). Let be a residuated lattice. Every prime ideal of second kind of is also a prime ideal. If is an MTL-algebra, then prime ideal of and prime ideal of second kind of are equivalent.
Let us recall the notion of maximal ideal.
Definition 8. (see [20]). Let be a proper ideal of a residuated lattice . is said to be a maximal ideal, if, for any ideal of , implies that or .
Proposition 2 (see [10]). Let be an ideal of . If is a maximal ideal of , then it is a prime ideal of .
Theorem 3 (see [10]). Let be a residuated lattice. If is an ideal of and is a filter of the lattice such that , then there is a prime ideal of such that and .
3. Fuzzy Ideal Generated by a Fuzzy Subset of Residuated Lattice
Let be a residuated lattice.
We first recall some definitions and properties of fuzzy subset.
Definition 9. (see [11]). Let be a nonempty set.(i)A map is called a fuzzy subset of (ii)A fuzzy subset of is called proper if it is not a constant map(iii)If is a fuzzy subset of and , then is called the -cut set of
Definition 10. (see [8]). Let be a fuzzy subset of . is a fuzzy ideal of , if it satisfies the following conditions: (FI1) for any , if , then (FI2) for any ,
Example 4. Let be a lattice defined by the Hass diagram of Figure 4.
Define and as follows (Table 1):
is a residuated lattice. The fuzzy subset of , defined byis a fuzzy ideal of .
Definition 11. (see [6, 13]). Let be a fuzzy subset of .(1) is a fuzzy filter of lattice, if it satisfies the following conditions:(i)for any , if , then (ii)for any , (2) is a fuzzy filter of residuated lattice, if it satisfies the following conditions:(i)for any , if , then (ii)for any , Note that any fuzzy filter of residuated lattice is a fuzzy filter of lattice ; but the converse is not always true.
Example 5. Let us consider the residuated lattice defined in Example 3. The fuzzy subset of , defined byis a fuzzy filter of lattice but it is not a filter of a residuated lattice because .
Let be two fuzzy subsets of . We define the order relation by , for all and, for any family of fuzzy ideals (or fuzzy filters) of A, and .
Let and denote, respectively, the set of fuzzy ideals of residuated lattice and the set of fuzzy filters of lattice .
Proposition 3. Let be a family of fuzzy ideals of a residuated lattice . Then is a fuzzy ideal of .
Proof. Let such that . We have ; then .
Moreover, for all , we have ; then . Thus, is a fuzzy ideal of .
In general, of fuzzy ideals of is not always a fuzzy ideal of . Indeed, this following example shows it.
Example 6. Let us consider the fuzzy ideals,of residuated lattice defined in Example 4. For , the fuzzy subset of , we havewhich is not a fuzzy ideal of because .
Let ; we define by , for all .
Proposition 4 (see [7]). Let be a fuzzy ideal of and such that . Then the fuzzy subset of is a fuzzy ideal of .
Theorem 4 (see [8]). Let be a fuzzy subset of . is a fuzzy ideal of if and only if, for each , implies that is an ideal of .
Notation 1. In the remainder of this paper, the following map will be very useful. Let be a nonempty subset of and such that . Define the map as follows: .
Proposition 5 (see [8]). is a proper fuzzy ideal of if and only if is a proper ideal of .
Definition 12. Let be a fuzzy subset of . A fuzzy ideal of is said to be generated by if and, for any fuzzy ideal of , if , then . The fuzzy ideal generated by will be denoted by .
Let be a fuzzy subset of . For all and ; then .
Theorem 5. Let be a fuzzy subset of . Then the fuzzy subset of defined by , for all , is the fuzzy ideal generated by .
Proof. Let . Suppose that ; take , for all and . We have ; then there exists such that and , for all . That is, and , for all . Then , for all , which implies that . Thus, . If , then, for all , ; therefore, for all , , which implies that ; that is, . Thus, . By Proposition 3, is an ideal of . If , then , where or . Therefore, by Theorem 4, is a fuzzy ideal of .
Let ; we have , where ; thus, ; that is, . Therefore, ; that is, .
Let be a fuzzy ideal of such that and . If , then . Suppose that . Then ; that is, for all , . Since , . Hence, .
Example 7. Let us consider the residuated lattice defined in Example 4. Let us consider the fuzzy subset of defined byLet ; if , then . If , then and . If , then . If , then . If , then . Therefore, we have Thus,
4. Fuzzy Prime Ideal Theorem
Definition 13. A proper fuzzy ideal of a residuated lattice is said to be fuzzy prime if or , for any .
Proposition 6. Let be a proper fuzzy subset of a residuated lattice . is a fuzzy prime ideal of if and only if, for any , if , then is a prime ideal of .
Proof. Let be a proper fuzzy subset of . Suppose that is a fuzzy prime ideal. Let such that . By Theorem 4, is an ideal of . Let such that ; that is, ; then or ; that is, or . Therefore, is a prime ideal of .
Conversely, suppose that, for all , if is nontrivial, then is a prime ideal. By Theorem 4, is a fuzzy ideal of . Let . Take ; we have or because if it is not, then , which is absurd ( is proper); that is, or ; that is, or . Thus, . Moreover, because ; then by hypothesis is a prime ideal and or ; that is, or . In addition, and because is a fuzzy ideal. Therefore, or .
If is a De Morgan residuated lattice, then is a fuzzy prime ideal of if or , for any .
Definition 14. A fuzzy ideal of a residuated lattice is said to be fuzzy prime of the second kind if it is nonconstant and or for any .
Example 8. Let be the residuated lattice defined in Example 4. Then, , defined byis a fuzzy prime ideal of the second kind of .
Lemma 1. Let be a residuated lattice. For all , we have and .
Proof. Let . We have by property (P4) of Theorem 1. Since , then . Thus, . In the same way, we show that .
Theorem 6. Any fuzzy prime ideal of the second kind of residuated lattice is a fuzzy prime ideal of . If is an MTL-algebra, then the notions of fuzzy prime ideal of and fuzzy prime ideal of the second kind of are equivalent.
Proof. Let be a fuzzy prime ideal of the second kind of . Let ; we have or . By Lemma 1, and ; then and . If , then . From the fact that is a fuzzy ideal, we have ; then . Identically, if , then . In conclusion, is a fuzzy prime ideal of .
Assume that is an MTL-algebra and let be a fuzzy prime ideal of . Then, satisfies the prelinearity condition; that is, for any , we have ; that is, ; then . Therefore, or , by hypothesis. Thus, is a fuzzy prime ideal of the second kind of .
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 lattice defined in Example 3 and let be the fuzzy ideal of defined by is a fuzzy prime ideal of which is not a fuzzy prime ideal of the second kind of , because and .
Definition 15. A proper fuzzy ideal of a residuated lattice is prime fuzzy if, for fuzzy ideals and of , implies that or .
Example 10. Let us consider the residuated lattice defined in Example 3.
Consider that the fuzzy subset , defined byis a prime fuzzy ideal of .
Remark 2. A fuzzy prime ideal of is not necessarily a prime fuzzy ideal of as the following example shows.
Example 11. Let us consider the residuated lattice defined in Example 3. Let be a fuzzy subset of defined by is a fuzzy prime ideal of . Let and be two fuzzy ideals of defined byWe haveThen , , and ; therefore, is not a prime fuzzy ideal of .
Definition 16. Let be a proper fuzzy ideal of . is called fuzzy maximal ideal if, for all , implies that is a maximal ideal of .
Example 12. Let us consider the residuated lattice defined in Example 3 and let be a fuzzy subset of defined by is a fuzzy maximal ideal of .
Proposition 7. Let be a fuzzy ideal of . If is a fuzzy maximal ideal of , then is a fuzzy prime ideal of .
Proof. Let be a fuzzy maximal ideal of . Then, for all such that , is a maximal ideal of . By Proposition 2, is a prime ideal of for all such that . Thus, according to Proposition 6, is a fuzzy prime ideal of .
The converse of the above proposition is not true. Let us consider the residuated lattice defined in Example 3 and let be a fuzzy subset of defined by is a fuzzy prime ideal of , which is not a fuzzy maximal ideal of because is not maximal.
Lemma 2. Let , let be a fuzzy ideal of a residuated lattice , and let be a fuzzy filter of the lattice . Then and are, respectively, the ideal of the residuated lattice and the filter of the lattice .
Proof. Let and such that . Then ; that is, . Let . We have . Since is a fuzzy ideal of , . Therefore, because . Thus, . Hence, is an ideal of .
Let and such that . Then ; that is, . Let ; we have . Therefore, . Thus, is a filter of the lattice .
Theorem 7. Let , let be a fuzzy ideal of a residuated lattice , and let be a fuzzy filter of the lattice such that . Then there exists a fuzzy prime ideal of such that and .
Proof. Let and . By Lemma 2, is an ideal of the residuated lattice and is a filter of the lattice .
Let us show that . Suppose that there exists . Then and . Therefore, ; this is a contradiction because, by hypothesis, . Thus, .
Using the fact that is an ideal of the residuated lattice , is a filter of lattice and and, by Theorem 4, there exists a prime ideal of the residuated lattice such that and . LetWe have ; then, by Proposition 5, is a fuzzy ideal of . If , then or ; that is, or ; therefore, or . If , then because . In addition, is a fuzzy ideal of and, for all , ; then, . Therefore, or , since and . Thus, is a fuzzy prime ideal of .
Let ; if , then . Else, ; then and , because ; in this case, . Thus, .
Let ; if , then and (because ). Therefore, . If , then . Thus, .
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.
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.