Abstract
This paper mainly focuses on building the -fuzzy filter theory of residuated multilattices. Firstly, we introduce the concepts of -fuzzy filter and -fuzzy deductive system of residuated multilattices. Then, we highlight their properties and show how they are linked. Secondly, we introduce the concept of prime -fuzzy filter and propose some illustrative examples. Then, we bring out their properties and show how they are related to the concept of -fuzzy prime filter. Thirdly, we characterize -fuzzy maximal filter and maximal -fuzzy filter by atoms and coatoms. In the case where is a distributive lattice, we prove that maximal -fuzzy filters are prime. Finally, we are interested in -fuzzy cosets of an -fuzzy filter, and we prove that the set of all -fuzzy cosets of any -fuzzy filter of a residuated multilattice is a residuated multilattice.
1. Introduction
The concepts of ordered and algebraic multilattice were introduced by Benado [1]. A multilattice is an algebraic structure in which the restrictions imposed on a complete lattice, namely, the “existence of least upper bounds and greatest lower bounds,” are relaxed to the “existence of minimal upper bounds and maximal lower bounds.” Cabrera et al. [2] introduced the notion of residuated multilattice as a partially ordered commutative residuated integral monoid (briefly pocrim) whose poset is a multilattice [3]. Therefore, residuated multilattices generalize both residuated lattices and multilattices.
Also, in 1965, Zadeh [4] introduced the notion of fuzzy subset of a nonempty set as a function from into the unit interval to describe, study, and formulate mathematically those objects which are not well defined. Many authors have investigated fuzzy algebraic notions taking the linearly ordered set to be the set of degrees of membership. However, as Goguen [5] pointed out, in some situations, the structure of a complete bounded lattice can be a suitable set of truth values.
In the same way, Tonga [6] studied the notion of maximal -fuzzy filters and -fuzzy maximal filters of bounded lattices. He showed that any maximal -fuzzy is also an -fuzzy maximal and gave characterizations of these. Following him, Kadji et al. [7] introduced the notions of -fuzzy filter and -fuzzy prime (and maximal) filter of residuated lattices and gave some characterizations of these notions. However, these different notions have not yet been explored in the residuated multilattice. In this paper, we introduce the notion of -fuzzy filter and -fuzzy coset of an -fuzzy filter in residuated multilattices.
The paper is organized as follows: In Section 1, we recall some basic notions on residuated multilattice and fix the different notations. In Section 2, we study the notion of -fuzzy filters with some examples and their properties. Section 3 is devoted to the notions of -fuzzy prime filter and prime -fuzzy filter. We give a complete characterization and several properties of -fuzzy prime filter and prime -fuzzy filter. In Section 4, we study -fuzzy maximal filters and maximal -fuzzy filters, and the relationships existing between these different classes of -fuzzy filters will also be given. In the last section, we will study the notion of cosets of a -fuzzy filter; we have shown that the set of cosets of a -fuzzy filter is also a residuated multilattice.
2. Preliminaries and Notations
In this preliminary section, we recall some essential facts about filters, congruences, and homomorphism of residuated multilattices; the reader is expected to refer to [2, 8–10] or [11] for details.
Given a partially ordered set and , a multisupremum of is a minimal element of the set of upper bounds of , and denotes the set of multisuprema of . Dually, we define the multi-infima, which will be denoted as .
Definition 1 (see [12]). A poset is an ordered multilattice if it satisfies that, for all with and , there exist such that and its dual version for . A multilattice is said to be full if and , for all .
We will write to denote and to denote the set .
In the following, we recall the comparability properties, which have an important role in multilattice theory.
Proposition 1 (see [12]). In any full multilattice , the following properties (called comparability laws) are satisfied:(C1) implies that and .(C2) implies that and .(C3)( and ) implies that , where .
Proposition 2. In a full multilattice , we have that, for all ,(i)if and , then there exists such that ;(ii)if and , then there exists such that .
Proof. (i)Let such that . We have , and then there exists such that and . Let such that , , and . We have and ; then, . Therefore, and because . Thus, .(ii)The proof is similar to .
Lemma 1. Let be a full multilattice and , three elements in . For all and , if , then .
Proof. Let such that and . Suppose that . We have . Let such that and . We have and . Then, . Since and , we obtain . Thus, .
Definition 2 (see [2]). A tuple is said to be a partially ordered commutative residuated integral monoid, briefly a pocrim, if the following properties hold:(1) is a commutative monoid with neutral element .(2) is a poset with maximum .(3)The operations and satisfy the adjointness condition; that is, if and only if , for every .We now recall the following useful conditions that hold in a pocrim , for all , which will be used:(P1) (i.e., and ).(P2) and .(P3)if , then , , and .(P4).(P5).(P6).(P7) and .
Definition 3 (see [2]). A residuated multilattice is a pocrim whose underlying poset is a multilattice. If, in addition, there exists a bottom element, the residuated multilattice is said to be bounded.
Example 1. Let be the multilattice described in Figure 1.
Let the subsets and . We define and as follows:andIt is a routine calculation that is a bounded residuated multilattice.
Lemma 2 (see [2]). Every residuated multilattice is full.
Given residuated multilattice , , and can be extended to as follows: for all , , , , and . Particularly, for all , , and will stand for , and , respectively.
Proposition 3 (see [2]). In a residuated multilattice , the following inclusions hold for all :(i).(ii).Particularly, and .
Let be a poset; the set of upper bounds of will be denoted by , and dually denotes the set of lower bounds. Similarly, for , upper closure isupper closure is, and lower closure is .
Proposition 4 (see [2]). [2] In a residuated multilattice , the following conditions hold for all :(1).(2).(3).(4).
Definition 4. A complete lattice is said to be a completely meet-distributive lattice if, for all and any family of , we have .
Definition 5. Let be a lattice and .(i) is said to be -prime if , and for any , implies either or .(ii) is said to be an atom of if there is no such that .(iii) is said to be a coatom of if there is no such that .Due to the combination of two structures, multilattice and pocrim, for any residuated multilattice, we have the notion of filter of pocrim (p-filter), the notion of filter of multilattice (m-filter), and the notion of filter of residuated multilattice (filter).
Definition 6 (see [2]). Given a pocrim, a non empty subset is a p-filter of if the following conditions hold:(i)If , then .(ii)If and , then . is a deductive system if it satisfies the following:(1).(2) and imply , for all .
Proposition 5. In the pocrim, the notions of p-filters and deductive systems are equivalent.
Proof. Let be a pocrim and . ) Suppose that is a p-filter. We have ; then, there exists such that . Moreover, ; therefore, . Let such that and . Then, we have. Therefore, . Thus, is a deductive system. ) Suppose that is a deductive system. Let such that and . We have and ,then by hypothesis.Let. Since , therefore, .
Thus is a p-filter.
Definition 7 (see [8]). Let be a multilattice. A nonempty set is said to be an m-filter if the following conditions hold: for all ,(M1) implies (M2) implies , for all (M3)If , then .
Definition 8 (see [2]). Let be a residuated multilattice. A nonempty set is said to be filter, if it is a deductive system and the following condition holds: implies and .
A filter of a residuated multilattice is said to be proper if ; it is non trivial if.
The existing relation between filter and m-filter is given by the following theorem.
Theorem 1 (see [2]). Let be a residuated multilattice and be a deductive system of ; is a filter if and only if(1) is an m-filter for all (2)for all , if , then (3)for all , if , then
Definition 9 (see [2]). Let be a residuated multilattice. A deductive system is said to be consistent if, for all , the following conditions hold:(i)If , then .(ii)If , then .
Definition 10. Let be a proper filter of residuated multilattice .
is a prime filter of if, for all , implies that or .
Let be a binary relation on and ; then, denotes that, for all , there exists such that , and for all , there exists such that .
Definition 11 (see [8]). Let be a multilattice; a congruence on is any equivalence relation such that if , then and for all .
Theorem 2 (see [8]). Let be a multilattice and be a binary relation on M. Then, is a congruence relation if and only if the following conditions hold:(i) is reflexive.(ii) if and only if there exist and with .(iii)If with and , then .(iv)If with , then and .
Definition 12 (see [12]). Let be a residuated multilattice; a congruence on is any equivalence relation such that if , then ; ; ; ; and for all .
The concept of homomorphism between hyperstructures [13–15] has been considered from different points of view in the literature. The most used definition is that introduced originally by Benado [1] in the framework of multilattice.
Definition 13 (see [8]). A map between multilattices is said to be a homomorphism if and for all .
Definition 14 (see [2]). Let be a map between residuated multilattices; is said to be a homomorphism if is a multilattice homomorphism and also and for all .
3. -Fuzzy Filters of Residuated Multilattices
Let be a bounded residuated multilattice and be a complete lattice. We will denote by and the induced order relation defined on and , respectively.
Definition 15 (see [6, 7]). Let be a nonempty set.(i)A map is called an -fuzzy subset of .(ii)An -fuzzy subset of is called proper if it is not a constant map.(iii)If is an -fuzzy subset of and , then is called the -cut set of .The notion of -fuzzy filter of a pocrim denoted by -fuzzy p-filter is defined as follows.
Definition 16. Given a pocrim. An-fuzzy subset of is an -fuzzy p-filter of if the following conditions hold for all :(i)If , then .(ii).
Example 2. Let . is a complete lattice.
Let with and , but is incomparable with and . The Hasse diagram of is given in Figure 2.
Then, is a pocrim with respect to the following operations (Tables 1 and 2).
The map defined byis an -fuzzy p-filter of . Since and are incomparable, then is not a chain.
The following result characterizes the -fuzzy p-filter of pocrim.
Proposition 6 (see [7]). Let be an -fuzzy subset of a pocrim . is an -fuzzy p-filter of if and only if, for all , is either empty or p-filter of .
Definition 17. Given a pocrim, an -fuzzy subset of is an -fuzzy deductive system of if it satisfies the following:(1), for all .(2), for all .The following proposition shows that, in a pocrim, the notions of -fuzzy p-filter and -fuzzy deductive system are equivalent.
Proposition 7 (see [7]). Let be a bounded pocrim. An -fuzzy subset of is an -fuzzy p-filter of if and only if it is an -fuzzy deductive system.
Definition 18. Let be a multilattice. An -fuzzy subset of is an -fuzzy m-filter if it satisfies the following conditions: for all ,(FM1)if , then ;(FM2), ;(FM3), .
Example 3. Let with , but and are incomparable. The Hasse diagram of is given in Figure 3.
Then, is a complete lattice. Let us consider the multilattice whose Hasse diagram is given in Figure 1. The map defined byis an -fuzzy m-filter of . We observe that is not a chain because and are incomparable.
Remark 1. If is an -fuzzy m-filter of a multilattice, then, for all , , for all . Indeed, if, then . Therefore, ; thus, .
Proposition 8. Let be a multilattice. An -fuzzy subset of is an -fuzzy m-filter if and only if, for all , the level set is either empty or m-filter.
Proof. ) Suppose that is an -fuzzy m-filter. Let such that . We have to prove that is an m-filter.(M1)Let and . We have because and . Therefore, . Thus, .(M2)Let , , and . We have , because ; then, . Thus, . (M3)Let . Suppose that . There exists such that ; that is, . Let . We have because is an -fuzzy m-filter; then, ; that is, . Thus, . ) Suppose that, for all , or is an m-filter. Let us show that is an -fuzzy m-filter.(FM1)Let such that ; that is, . We have ; then, is an m-filter; therefore, ; that is, . Thus, .(FM2)Let and . We have ; then, , where . Therefore, ; thus, . In addition, ; then, by (FM1), ; therefore, . Thus, .(FM3)Let . We have and ; then, and are an m-filter. We have and ; that is, and . Then, ; therefore, and ; that is, and . Thus, .
Definition 19. An -fuzzy subset of is said to be an -fuzzy filter of if it is an -fuzzy deductive system of , and for all , , for all .
Example 4. Let be a complete lattice whose Hasse diagram is given in Figure 3. Let be the residuated multilattice defined in Example 1. The map defined byis an -fuzzy filter of . We have and are incomparable; then, is not a chain.
The map defined byis an -fuzzy p-filter of which is not an -fuzzy filter because and .
Remark 2. (i)If we replace the complete lattice by the lattice , then this definition of -fuzzy filter of residuated multilattice will coincide with the one given by L. N. Maffeu Nzoda et al. [10].(ii)The -fuzzy filter defined in Example 5 is an -fuzzy m-filter.(iii)The -fuzzy p-filter defined in Example 5 is not an -fuzzy m-filter because and .The following result characterizes the -fuzzy filter of residuated multilattice in terms of its level subset.
Proposition 9. Let be an -fuzzy subset of . is an -fuzzy filter of if and only if, for each , implies that it is a filter of .
Proof. ) Suppose that is an -fuzzy filter of . Let such that . By Proposition 6, is a deductive system because is an -fuzzy deductive system. Let such that . Let and . We have and . Therefore, and ; thus, and . Hence, is a filter of . ) Suppose that, for all , is either empty or a filter of . By Proposition 6, is an -fuzzy deductive system of . Let , , and . We have and . Then, and , because is an -fuzzy deductive system of .Let . We have . Then, and , because is a filter by hypothesis. Thus, and ; that is, and .
Hence, and . Thus, is an -fuzzy filter.
Proposition 10. Let be a homomorphism between residuated multilattices and a -fuzzy filter of . Then, the inverse image of denoted by is a -fuzzy filter of , where for all , .
Proof. Let such that . We have ; that is, ; then, . Therefore, because is an -fuzzy filter. Thus, .
Let . We have ; then, . Thus, .
Let , , and . We have . Therefore, and , because is an -fuzzy filter. Since is a residuated multilattices homomorphism, we have and . Therefore, and ; that is, .
Thus, is an -fuzzy filter of .
An -fuzzy filter of is also an -fuzzy m-filter of , but the converse is not always verified. Indeed, the -fuzzy m-filter of Example 3 is not an -fuzzy p-filter of the residuated multilattice of Example 1 because . Therefore, it is not an -fuzzy filter. The following theorem gives the link between the -fuzzy p-filter, -fuzzy m-filter, and -fuzzy filter of a residuated multilattice.
Theorem 3. Let be an -fuzzy subset of . is an -fuzzy filter of if and only if(i) is an -fuzzy deductive system of ;(ii) is an -fuzzy m-filter of ;(iii)for all , , for all , where .
Proof. ) Suppose that is an -fuzzy filter of .(i)By definition of -fuzzy filter, is an -fuzzy deductive system of .(ii)Let us show that is an -fuzzy m-filter of .(FM1)Let . If , then because is a deductive system.(FM2)Let . Given that is an -fuzzy deductive system, we have because . By (FM1), , because . Hence, .(FM3)Let . If is a singleton, then and . Suppose that is not a singleton and . By Proposition 2, there exists such that . Since is an -fuzzy deductive system, we have because and . From , we obtain because and . Therefore, . Similarly, we prove that . Thus, . Hence, is an -fuzzy m-filter of .(iii)Let . Let us prove that . If is a singleton, then and . Suppose that . Then, by Proposition 2 (i), there exists such that . Since , we have because and . We have because , and then . Similarly, we prove that .Let . The proof of is similar to the above by using Proposition 2 (ii).
) Suppose that , , and are satisfied. Since is an -fuzzy deductive system, let . We have to prove that, for all and , .
By Proposition 3, we have ; then, there exist and such that and .
Let . If , then . If not, ; that is, there exists , such that and . Since , we have by the hypothesis and because is an -fuzzy deductive system. By property of the pocrim, we have , and then . Since , we have . Therefore, . We have ; then, . Hence, .
Similarly, we prove that, for all , .
Since is an -fuzzy deductive system by hypothesis, we conclude that is an -fuzzy filter.
Definition 20. An -fuzzy deductive system of is said to be consistent if, for all , the following conditions hold:(i), for all .(ii), for all .
Example 5. Let be a complete lattice whose Hasse diagram is given in Figure 3. Let be the residuated multilattice defined in Example 1. The only consistent -fuzzy deductive systems of this residuated multilattice are the constant maps of to .
Proposition 11. Every consistent -fuzzy deductive system of is an -fuzzy filter of .
Proof. Let be a consistent -fuzzy deductive system of . Let , , and . We have and , and then and because . We have , and then , . Therefore, . Thus, is an -fuzzy filter of .
The converse of this proposition is not always true, and the following example is an illustration.
Example 6. The -fuzzy filter of Example 5 is not consistent because , where .
In the rest of this paper, we will use the following -fuzzy subset of . Let be a nonempty subset of and such that . Define the map as follows: for all ,
Remark 3. It is easy to see that, for all nonempty subset of , the -fuzzy subset of is a proper -fuzzy filter of if and only if is a proper filter of .
Proposition 12. Let and be two -fuzzy filters of . Then, the -fuzzy subset of defined by , for all , is an -fuzzy filter of .
Proof. Let . Let such that . We have and , and then ; that is, .
Let . We haveHence, is an -fuzzy deductive system.
Let , , and . We have and . Then, and , because and are -fuzzy filters of .
Thus, is an -fuzzy filter of .
In general, given and two -fuzzy filters of , the -fuzzy subset defined by , for all , is not always an -fuzzy filter as shown in the following example.
Example 7. Let be a complete lattice whose Hasse diagram is given in Figure 3. Let be a residuated multilattice defined in Example 1. The map , defined byis an -fuzzy filter of . Consider the -fuzzy filter of defined in Example 5. We haveThen, is not an -fuzzy filter of this residuated multilattice because .
Let denote the set of all -fuzzy filters of . On this set, we define the order relation as follows: for all , if and only if , for all .
Definition 21. Let be an -fuzzy subset of . An -fuzzy filter of is said to be generated by if , and for any -fuzzy filter of if , then ; that is, is the smallest -fuzzy filter of that contains . The -fuzzy filter generated by will be denoted by .
For all , we define. Let and be an -fuzzy subset of defined by . It is easy to show that is a bounded lattice.
4. -Fuzzy Prime Filter of Residuated Multilattice
This section is devoted to the notions of -fuzzy prime filter and prime -fuzzy filter of a residuated multilattice. Their characterizations will be given as well as the relation which links them.
Definition 22. Let be a proper -fuzzy filter of . is an -fuzzy prime filter if, for all , is a prime filter of , whenever .
Example 8. Let us consider the residuated multilattice defined in Example 1 and the lattice . The map defined byis an -fuzzy prime filter of .
Remark 4. Let be a nonempty subset of . The -fuzzy subset of is an -fuzzy prime filter of if and only if is a prime filter of .
Proposition 13. Let be a proper -fuzzy filter of . If is an -fuzzy prime filter, then is a chain.
Proof. Let be an -fuzzy prime filter of . Let and . If , then ; that is, .
Suppose that . Because , we have . Hence, is a prime filter. Let , and we have ; then, ; therefore, ; that is, . Thus, . Since is a prime filter, then or ; that is, or . Therefore, or ; hence, is a chain.
The following result characterizes an -fuzzy prime filter of residuated multilattice.
Theorem 4. Let be a proper -fuzzy filter of . is an -fuzzy prime filter of if and only if is a chain and for all .
Proof. Suppose that is an -fuzzy prime filter of . By Proposition 13, is a chain. Let . By hypothesis, is a -fuzzy m-filter; then, for all , ; thus, . In addition, let . We have , for all . Then, . If , then and . If not, is a prime filter of ; therefore, or . In any case, we have or . Because , we have . Hence, .
Conversely, suppose that is a chain, and for all , . Let such that . Let such that and . We have for all , and then . Since is a chain and , we have ; that is, . Thus, is a prime filter.
Now we introduce the notion of prime -fuzzy filter of residuated multilattice , which is a prime element in the lattice of -fuzzy filters of .
Definition 23. Let be a proper -fuzzy filter of . is a prime -fuzzy filter of if, for all , implies or .
Example 9. Let us consider the residuated multilattice defined in Example 1 and the lattice . The map defined byis a prime -fuzzy filter of .
Lemma 3. Let be a proper -fuzzy filter of . If is a prime -fuzzy filter of , then the following conditions hold:(1).(2).(3) is an -prime.
Proof. (1)Suppose that . Let and be two -fuzzy subsets of defined by and for all . Then, is an -fuzzy filter by Proposition 8 and Proposition 9. is an -fuzzy filter because it is a constant map. We have For all , . Therefore, , for all . If , then ; that is, , for all . Hence, . Since is a proper, there exists , such that . Hence, ; that is, . In addition, , because ; that is, . Therefore, , , and ; this contradicts the fact that is a prime -fuzzy filter of . Hence, .(2)Let such that . Suppose that ; that is, . Let and be two -fuzzy subsets of defined by and , for all . Then, is an -fuzzy filter by Remark 3 and 2.4; is an -fuzzy filter because it is a constant map. We have It is easy to show that . Since , we have and ; this contradicts the fact that is a prime -fuzzy filter of . Thus, for all ; hence, .(3)Let such that . Let and be two -fuzzy subsets of defined by and , for all . We have for all ; then, . Since and are -fuzzy filters of and is a prime -fuzzy filter, we have or . Therefore, or ; hence, is an -prime.It is clear that an -fuzzy prime filter is not always a prime -fuzzy filter because the example of -fuzzy prime filter defined in Example 8 is not a prime -fuzzy filter. In addition, a prime -fuzzy filter is not always an -fuzzy prime filter according to Example 9. The following theorem gives the relation existing between -fuzzy prime filter and prime -fuzzy filter of residuated multilattice.
Remark 5. By Proposition 13 and Remark 4, a prime -fuzzy filter of is an -fuzzy prime filter of if and only if is a prime filter of .
Lemma 4. Let be a filter of , be an -prime element of , and be an -fuzzy subset of defined byThen, is a prime -fuzzy filter of .
Proof. We have , and is a filter, then by Remark 3, and is an -fuzzy filter of . Let and be two -fuzzy filters of such that . We have , and then .
If , then and because .
Else, , and then or because is an -prime element of . In any case, we have or . Thus, is a prime -fuzzy filter of .
The following result characterizes prime -fuzzy filter of .
Theorem 5. Let be a proper -fuzzy filter of . is a prime -fuzzy filter of if and only if and is an -prime element of .
Proof. Let be a proper -fuzzy filter of .
Suppose that is a prime -fuzzy filter of , and then by Lemma 3, and is an -prime element of .
Conversely, suppose that and is an -prime element of . We haveBy Proposition 9 and Lemma 4, is a prime -fuzzy filter of .
5. -Fuzzy Maximal Filter of Residuated Multilattices
Definition 24. Let be a proper filter of . is a maximal filter of if, for all filter of , implies that or .
Definition 25. A proper -fuzzy filter of is called -fuzzy maximal filter of if for each , when is nontrivial, then is a maximal filter of .
Example 10. Let be a complete lattice, where . Let be a residuated multilattice defined in Example 1. Consider the map defined by, for all , is a -fuzzy maximal filter of .
Theorem 6. Let be a proper -fuzzy filter of . is an -fuzzy maximal filter if and only if, for all , implies or is a maximal filter of .
Proof. Suppose that is an -fuzzy maximal filter. Let such that . Let and . If and are incomparable, then and are nontrivial filters. By hypothesis, and are maximal filters. Therefore, is a maximal filter. Else, if , then ; that is, . Suppose that ; then, ; that is, is a nontrivial filter. By hypothesis, is a maximal filter of .
Conversely, let us assume that, for all , implies or is a maximal filter of . Let such that is nontrivial; that is, . There exists such that , , and ; that is, , , and . Then, there exists such that and . Hence, by hypothesis, is a maximal filter of . Moreover, because . Since and is maximal, we have . Thus, is a maximal filter of .
Proposition 14. Let be an -fuzzy maximal filter of . In the ordered set , every element of is both an atom and a coatom.
Proof. Let .
Let us show that is an atom. Suppose that there exists such that . We have . Moreover, , and then by Theorem 6 and since , is a maximal filter. We have and ; this contradicts the fact that is a maximal filter. Then, there is no such that . Thus, is an atom of .
Similarly, we prove that is a coatom of .
Now we will define the maximal element of .
Definition 26. Let be a proper -fuzzy filter of . is a maximal -fuzzy filter of if for every -fuzzy filter of , implies that or .
Example 11. Let be a complete lattice, where . Let be a residuated multilattice defined in Example 1. Consider the map defined by is a maximal -fuzzy filter of .
Remark 6. In Example 10, the -fuzzy maximal filter of is not a maximal -fuzzy filter because we have , , and , where is a -fuzzy filter of defined in Example 11.
In the rest of this section, we assume that is a completely meet-distributive lattice.
Lemma 5. Let be an -fuzzy filter of and . Then, is an -fuzzy filter of .
Proof. Let . Let such that . We have , and then .
Let . We have because is a meet-distributive. Then, .
Let , , and . We have ; then, and . Thus, is an -fuzzy filter of .
Proposition 15. Let be a maximal -fuzzy filter of . Then, and is a coatom of .
Proof. Since is proper, . Let such that ; that is, there exists such that . By Lemma 5, is an -fuzzy filter, and . We have ; then, . Therefore, because is maximal. Hence, ; that is, . Therefore, .
Now let us show that is a coatom. Suppose that is not a coatom; then, there exists such that . Consider the map defined by, for all , is an -fuzzy filter, because where . We have , , and , then contradicting the maximality of . Thus, is a coatom in .
Proposition 16. If is a distributive lattice, then every maximal -fuzzy filter of is a prime -fuzzy filter of .
Proof. Let be a distributive lattice. Let be a maximal -fuzzy filter of . By Proposition 15, , where is a coatom of . By Theorem 5, we only have to prove that is an -prime element of . Let such that . Suppose that and ; that is, and . Then, and , because is a coatom. Therefore, , which contradicts the fact that is a coatom. Hence, is an-prime element of . Thus, is a prime -fuzzy filter of .
Since in a lattice an -prime element is not always a coatom, a prime -fuzzy filter is not always a maximal -fuzzy filter.
Lemma 6. If is a maximal -fuzzy filter of , then is a maximal filter of .
Proof. Let be a maximal -fuzzy filter of . Then, by Proposition 15, and is a coatom. Suppose that is not a maximal filter of ; then, there exists a proper filter of such that . Let be the map . Let . If , then ; that is, . Therefore, and . Else, if x ∊ H, then and . Hence, .
On the other hand, we have , and then . And since H ≠ M, we have . Hence, , , and . This contradicts the fact that μ is a maximal -fuzzy filter of . Thus, μ1 is a maximal filter of .
Remark 7. By Definition 25, it is clear that if is a completely meet-distributive lattice, then a maximal -fuzzy filter of a residuated multilattice is an -fuzzy maximal filter of .
Theorem 7. A proper -fuzzy filter μ of is a maximal -fuzzy filter if and only if , where G is a maximal filter of and α is coatom in L.
Proof. Suppose that μ is a maximal -fuzzy filter of . By Proposition 15 and Lemma 6, . , is a coatom, and G = μ1 is a maximal filter. We have μ(x) = 1 or μ(x) = α. If μ(x) = 1, then and . Else, if μ(x) = α, then and . Thus, .
Suppose that , where G is a maximal filter of and α is coatom in L. We haveThen, G = μ1.
Let θ be a proper -fuzzy filter of such that . If , then μ(x) = 1 and ; that is, . Else, if , then . Since α is a coatom, then or ; if , then and ; that is, , hence contradicting the maximality of G. Therefore, . Then, ; thus, μ is a maximal -fuzzy filter of .
6. -Fuzzy Cosets of an -Fuzzy Filter
Definition 27. Let be a residuated multilattice. Let μ be an -fuzzy filter of and . A -fuzzy coset of the -fuzzy filter μ is the -fuzzy subset μa of defined by , for all .
Let μ be an -fuzzy filter of and . Consider the relation defined on by, for all , if and only if .
Remark 8. Let μ be an -fuzzy filter of and such that . Then, is an equivalence relation on .
Proposition 17. Let μ be an -fuzzy filter of and , such that . Then, is an equivalence relation on compatible with and .
Proof. By Remark 8, is an equivalence relation on .
Let such that . By P6 and P7, we have . Then, . Similarly, we have . Therefore, . Thus, , and .
Theorem 8. Let μ be an -fuzzy filter of ; then, is a congruence relation on .
Proof. Let us show, by using Theorem 2, that is a congruence of multilattice . By Proposition 17, is an equivalence relation on M; then, (i) and (iii) of Theorem 2 are verified.
(ii) Let . Suppose that ; then, . For all and , we have ; then, . Since μ is an -fuzzy filter, and , for all and . By P5, we have for all and , ; then, . Thus, , for all and .
Conversely, assume that there exist and with . By hypothesis, . We have and ; then, ; that is, ; that is, . Similarly, we prove that . Therefore, .
(iv) Let such that and . Since , for all , and by definition of , we have .(i)Let us prove that, for all . Let . We have and ; then, there exists such that ; then, . We have ; then, and . Since μ is an -fuzzy filter, we have , since by Lemma 1. Hence, . Thus, for all , there exists such that . Let . Check such that . For all , we have ; then, or z and y are incomparable; that is, or . Suppose that there exists such that . Then, by Lemma 1, we have . Therefore, because and ; that is, . Moreover, we have; then, . In this case, take x = z. Suppose that, for all , ; that is, . If take x = y because is reflexive. If not, a and y are incomparable. Let ; we have ; then there exists such that . By Lemma 1, . We have and ; that is, . Since , we have because and μ is an -fuzzy filter of . Therefore, ; that is, . In this case, take .(ii)Similarly, we prove that , for all .Hence, is a congruence on multilattice . And by Proposition 17, is a congruence of pocrim M. Thus, is a congruence of residuated multilattice .
Lemma 7. Let μ be an -fuzzy filter of and . if and only if .
Proof. Let . If , then ; thus, .
Conversely, if , then ; that is, . Since μ is an -fuzzy filter, for all , , because . Therefore, . Similarly, we prove that . Thus,. We also have ; then for all , .
Remark 9. Let μ be an -fuzzy filter of , and let M/μ denote the set of all coset of μ; that is, .
For all , if μx = μ and μy = μb, then , , , and , where , for all . Therefore, are well defined on .
Thus, for any , we define ; ; ; and .
Proposition 18. Let μ be an -fuzzy filter of . For all , if and only if .
Proof. Let . Suppose that ; we have. Then, for all , ; that is, . Let ; we have ; then, . Let . Since μ is an -fuzzy filter, we have , because . We have thus proved that, for all , ; that is, . It follows that .
Similarly, we prove that if , then .
Let μ be an -fuzzy filter of . Let . We have if and only if , for all . Particularly, if , then ; that is, . It follows that if and only if . Therefore, the classical ordering relation on (M/μ) leads to equality. Consequently, we need to build a suitable order relation on (M/μ).
Proposition 19. Let μ be an -fuzzy filter of . The relation defined for all , if and only if is an ordering relation on (M/μ).
Proof. is easily seen to be reflexive and antisymmetric.
Let such that and . Then, for all , , , and . Let us show that ; that is, . Let . We have and ; then,. For all , , because . Also, we have for all , , because . Then, ; that is, . Hence, , for all ; that is, . Therefore, .
Thus, is an ordering relation on (M/μ).
Now, we have all the required properties to prove that the set of all -fuzzy cosets of an -fuzzy filter of a residuated multilattice is a residuated multilattice. First, we prove that it is a multilattice.
Proposition 20. Let μ be an -fuzzy filter of . is a multilattice.
Proof. Let such that and . Check such that . Let and . We have and by hypothesis. Since and , then there exists such that . Since and , then and . Therefore, . We have ; then, there exists such that ; that is, . We have ; then . Thus, .
Let ; we have . Also, ; then, . Therefore, for all , ; that is, . Thus, ; that is, . Take .
Similarly, we prove the dual.
We now prove that the multilattice of all -fuzzy cosets of an -fuzzy filter of a residuated multilattice is a residuated multilattice.
Proposition 21. Let μ be an -fuzzy filter of . Then, is a residuated multilattice.
Proof. We have that is an ordered multilattice by Proposition 20 and is a commutative monoid with neutral element by definition of .
Let . Suppose that ; that is, , for all . Hence, for all , because and μ is an -fuzzy filter of . Let ; we have , by property P4 of pocrim. Since , then . Therefore, ; that is, for all . Thus, ; that is, .
Suppose that ; then, for all , because . Let . We have . Since , we have . Hence, . Thus, ; that is, .
Define the map by , for all .
Theorem 9. Let μ be an -fuzzy filter of . Then,(i) is a surjective homomorphism of residuated multilattice;(ii);(iii), where is the set of all equivalence classes of the congruence relation .
Proof. Let μ be an -fuzzy filter of .(i)By Lemma 5.5, is a residuated multilattice. For all , we have ; ; ; ; and. Thus, by definition of , it is a surjective homomorphism of residuated multilattice.(ii) if and only if iff iff iff iff iff . Thus, .(iii)Let us define the map by . We have iff iff . Then, φ is well defined, and it is injective. φ is surjective by definition. Thus, .
7. Conclusion
We have introduced the notion of -fuzzy filter of residuated multilattice, which generalizes the work of Maffeu et al. [12] by replacing the closed unit interval [0, 1] by a complete lattice . Besides that, we have studied the notion of -fuzzy prime filter of residuated multilattice and prime -fuzzy filter. It appears that a prime -fuzzy filter μ is an -fuzzy prime if the 1-cut set of μ is a prime filter. The concept of maximal -fuzzy filter and -fuzzy maximal filter was defined; it follows that a maximal -fuzzy filter is also an -fuzzy maximal filter if is distributive, but the converse is not always verified. Finally, we introduce the notion of -fuzzy cosets of an -fuzzy filter, and we prove that the set of all -fuzzy cosets of an -fuzzy filter is also a residuated multilattice. For future work, we will study the notions of -fuzzy congruence and -fuzzy homomorphism of residuated multilattice.
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.