Abstract
Given a pseudoresiduated lattice and a lattice , we introduce and characterize the fuzzy versions of different -fold implicative (resp., obstinate, Boolean, normal, and extended involutive) filters of . Moreover, we study some relationships between these different types of fuzzy -fold filters.
1. Introduction
Uncertainty is present in almost every sphere of our daily life. Traditional mathematical tools are not sufficient to handle all practical problems in fields such as medical science, social science, engineering, and economics, involving uncertainty of various types. Zadeh, in 1965, came up with his remarkable theory of fuzzy sets for dealing with these types of uncertainties where conventional tools fail. His theory brought a grand paradigmatic change in mathematics. Later on, fuzzy logic became popular.
Fuzzy logic has become a useful tool for computer science to deal with uncertain information. Some domains of research and application of fuzzy logic are automation (in manufacture), instrumentation (captors, instruments of measure, and recognition of voice), judgment and conception (consultation, decision making, and timetable for traffic). Recently, an algorithm based on fuzzy logic is used for the detection of the cancer tumors [1].
The origin of residuated lattices lies in mathematical logic without contraction. Residuated lattices have been investigated by Krull [2], Dilworth [3], Balbes and Dwinger [4], and Pavelka [5]. Many algebraic structures have been proposed as the semantic aspects of logical systems. In 1958, Chang introduced the notion of MV-algebra for many-valued logic, and his completeness theorem (1959) stated the closed real unit interval as a standard model of this logic. Later on, BL-algebra was introduced by Hàjek [6] as a commutative generalization of MV-algebra. Filters correspond to sets of formulae closed with respect to Modus Ponens. So filters are important tools in studying the above algebras and the completeness of nonclassical logic. From the logical point of view, various filters correspond to various sets of provable formulae.
At present, there are two ways to generalize the existing types of filters: one is folding theory and the other is fuzzy set theory.
In the folding approach, Haveshki and Eslami introduced the notion of -fold implicative (resp., -fold positive implicative) BL-algebra [7]. Then, Turunen et al. in [8, 9] proved that -fold implicative BL-algebras are Gödel algebras, and -fold positive implicative BL-algebras are MV-algebras.
In the fuzzy approach, a lot of work has been done on fuzzy filters which are particular cases of fuzzy sets introduced by Zadeh [10] (a fuzzy subset of a set is a map , where is the closed unit interval of real numbers); most of the authors (see, e.g., [11–22]) use the above original definition of a fuzzy subset.
In the present work, we replace the closed real unit interval by an arbitrary lattice. So, a fuzzy subset of will be a map , where is the underlying set of a lattice. It follows from our characterizations of fuzzy filters that most of the known results with are easily proved and extended to more general cases. We also introduce some new notions and prove several new results in pseudoresiduated lattices.
The paper is organized as follows. In Section 2, some basic concepts and properties are recalled. In Section 3, we study the notions of -fuzzy filter. Section 4 is devoted to the notions of -fuzzy -fold obstinate filter and -fuzzy -fold Boolean filter. In Section 5, we study the notion of -fuzzy -fold implicative filter and show that any -fuzzy -fold Boolean filter is -fuzzy -fold implicative. In Section 6, the notions of -fuzzy -fold normal (resp., extended involutive, fantastic, and involutive) filter are given and we obtain that an -fuzzy filter is -fuzzy -fold Boolean if and only if it is -fuzzy -fold fantastic and -fuzzy -fold implicative.
2. Preliminaries
In this section, we recall some definitions and results which will be used in the sequel.
A pseudoresiduated lattice is a nonempty set with five binary operations , , , , and and two constants satisfying the following:L-1: is a bounded lattice.L-2: is a monoid.L-3: iff if and only if (pseudoresiduation).
A pseudo-RL monoid is a pseudoresiduated lattice which satisfies the following condition:L-4: (pseudodivisibility).
A pseudo-MTL algebra is a pseudoresiduated lattice which satisfies the following condition:L-5: (pseudoprelinearity).
A pseudo-BL-algebra is a pseudo-MTL-algebra which satisfies the pseudodivisibility.
A pseudo-MV-algebra is a pseudo-BL-algebra which satisfies the following condition:L-6:, where and .
Alternatively, a pseudo-MV-algebra can be defined as a pseudoresiduated lattice which satisfies the following condition:L-7: and .
A Heyting algebra is a pseudoresiduated lattice which satisfies the following condition:L-8:.
A Boolean algebra is a pseudoresiduated lattice which satisfies the following condition:L-9:.
Recall that, for any , , for all .
Proposition 1 (see [23]). For all , the following properties hold: (1) if and only if .(2); ; .(3); .(4)If then: , , , , , , and .(5); ; ; .(6); ; ; .(7); .(8); .(9); .(10); .(11); .(12); .(13); .(14); ; .
In the following, unless mentioned otherwise, any pseudoresiduated lattice will often be referred to by its support set and will be called a residuated lattice for short.
Definition 2. Let and be two residuated lattices. Then, a map is called a residuated lattice homomorphism if preserves all the operations, that is, if the following conditions are satisfied: (i) is a morphism of bounded lattices.(ii)For all , .(iii)For all , .(iv)For all , .
If is bijective, the homomorphism is called a residuated lattice isomorphism. In this case, we write .
Recall that a nonempty subset of is called a filter if it satisfies the following two conditions:(F1)For every , .(F2)For every , if and , then .
A filter is said to be proper if .
A filter is said to be commutative if, for all , iff .
Remark 3. Let be a commutative filter of . The binary relation , defined on by if and only if and , is congruence on . The quotient structure is also a pseudoresiduated lattice, where, for all ,Note that .
The following examples will be frequently used in the text.
Example 4. Let be a lattice such that , where and are incomparable. Define the operations , , and by the three tables in (2). Then is a residuated lattice which is not a pseudo-MTL-algebra, since . Consider and are the only proper filters of .
Example 5. Let be a lattice such that , , where and are incomparable. Define the operations , , and by the three tables in (3). is a residuated lattice which is not a pseudo-MTL-algebra, since . Consider; ; ; are the proper filters of .
and are commutative filters, but is not, since and .
We recall that a proper filter is said to be prime if, for all , implies or . So all filters of Example 5 are prime filters.
Proposition 6 (see [12]). The following conditions are equivalent for a filter :(i) is a prime filter.(ii)For any filter of , implies or .
Note. Similar to the case of commutative residuated lattices [24], we establish in [25] that(i)a filter is maximal if and only if, for any , implies that there are and natural numbers such that ;(ii)any proper filter can be extended to a prime filter , and any proper filter can be extended to a maximal filter . So, any maximal filter is prime.
3. Fuzzy Filter
3.1. Definitions and First Properties
Definition 7. Let be a residuated lattice and let be a lattice.
(i) A function is called an -fuzzy subset of .
For all , the set is called the -cut of .
(ii) An -fuzzy subset of is called an -fuzzy filter of if, for all , is either empty or a filter of .
For any , let denote the constant function with value . Clearly, is an -fuzzy filter of (called a constant fuzzy filter).
An -fuzzy filter is said to be proper if it is nonconstant.
From now on, an -fuzzy subset (resp., filter) will be called a fuzzy subset (resp., filter).
Example 8. Let be a proper filter of and such that . Define the map by , if , and , if .
It is easy to see that is a nonconstant fuzzy filter of .
The following result was established in [12].
Theorem 9. A fuzzy subset of is fuzzy filter of if and only if, for all ,(i),(ii) implies .
Proposition 10. The following assertions are equivalent for any fuzzy subset and any :(i) is a fuzzy filter.(ii) for all , and .(iii) for all , and .
Proof. Since , we have .
In addition, since , it follows that .
Let . If , then ; that is, . Now, since , it follows that .
It is similar to .
Let denote the set of all fuzzy filters of .
Proposition 11. For all and , a fuzzy filter, the following assertions hold:(1), and .(2).(3).(4), and .(5), and
Proof. Let be a fuzzy filter, and :(1)By Proposition 10, ). Similarly, we show that .(2)This follows from the fact that and .(3).(4)Since , we have ; so .(5)Note that .
Definition 12. A fuzzy filter is said to be a commutative fuzzy filter if, for all , if and only if .
It is easy to show the following result.
Lemma 13. is a commutative fuzzy filter of if and only if is a commutative filter of .
In particular, if is a filter of and in , then is a commutative fuzzy filter of if and only if is a commutative filter of .
A class of filters of is said to be closed under extension if, for all filters of such that , implies .
A class of fuzzy filters is said to be closed under extension if, for all such that and , implies .
3.2. Fuzzy Prime Filter
Definition 14. A nonconstant fuzzy filter is called a fuzzy prime filter if, for every , is a prime filter whenever it is proper.
Example 15. Let be a lattice:(i)Let be the residuated lattice of Example 5; then any fuzzy filter is fuzzy prime.(ii)Let be a filter of , and in ; then is a fuzzy prime filter if and only if is a prime filter of .
In fact, we have the following result.
Theorem 16. A nonconstant fuzzy filter is fuzzy prime if and only if is a chain, and for all .
Proof. Let be a nonconstant fuzzy filter of .
Assume that is fuzzy prime; then , so is or a prime filter; thus or ; this implies or ; so or .
Conversely, assume that the conclusion holds. Let such that is a proper filter, and such that . That is, ; since , we obtain or . So is a prime filter of .
4. Fuzzy -Fold Obstinate Filter
Definition 17. Let be a residuated lattice, and : (i)A proper filter is -fold obstinate if, for all , if and only if and .(ii) is said to be an -fold obstinate residuated lattice if, for all , .
Remark 18. (i) In Example 4, the filter is -fold obstinate for , but the filter is not, since and for .
(ii) If is an -fold obstinate filter and , then ; so for all , showing that is a maximal filter.
It is easy to show the following result.
Proposition 19. Let : (i)A commutative filter is -fold obstinate if and only if is -fold obstinate.(ii) is -fold obstinate if and only if is an -fold obstinate filter.
Definition 20. Let .
A proper fuzzy filter of is said to be fuzzy -fold obstinate if, for all , is an -fold obstinate filter of whenever it is proper.
Example 21. Let be the residuated lattice of Example 4 and a lattice. Then, for all in , the fuzzy filter is a fuzzy -fold obstinate filter for , but the fuzzy filter is not, since and for .
Theorem 22. A fuzzy filter is fuzzy -fold obstinate if and only if, for any , if and only if .
Proof. We first note that, for a proper filter and , (resp., ) implies .
implies that ; so , and the latter is a proper filter as is proper; so , and .
Let such that is a proper filter of . Let such that ; since , we have ; so , and . Thus is an -fold obstinate filter of .
Remark 23. Since and , any fuzzy -fold obstinate filter is fuzzy -fold obstinate.
Corollary 24. A proper filter of is -fold obstinate if and only if, for all in , is a fuzzy -fold obstinate filter of . In particular, is -fold obstinate if and only if is a fuzzy -fold obstinate filter.
Definition 25 (see [25]). (i) A filter of is -fold Boolean if and for all .
(ii) A residuated lattice is -fold Boolean if for all .
Example 26. Let be a lattice such that . Define the operations , , and by the three tables in (4). Then is a residuated lattice which is not a pseudo-BL-algebra, since . Consider is the only proper filter of and it is an -fold Boolean filter for .
is an -fold Boolean residuated lattice for all .
One can observe that is not a Boolean algebra since
Definition 27. A fuzzy filter is fuzzy -fold Boolean if, for all , .
In particular, a fuzzy 1-fold Boolean filter is a fuzzy Boolean filter [12].
Proposition 28. The following conditions are equivalent for any fuzzy filter and any :(i) is fuzzy -fold Boolean.(ii)For all , .(iii)For all , .(iv)For all , .(v)For all , .(vi)For all , implies is an -fold Boolean filter.
Proof. Suppose that is fuzzy -fold Boolean. Let . Since , we have , so , as is fuzzy -fold Boolean. Similarly, and then .
Let and ; and we must show that . Since , we have and then . Hence and ; from this, we have , by (ii) and . Similarly, we can prove that .
By the same arguments as for .
Let . By Proposition 1(4), we have and then ; this implies (a).
Similarly, we show that (b).
Now the result follows from (a), (b), and (iii)
Let in conditions .
It is similar to the proof of .
It follows from Definition 27.
Theorem 29. The following assertions are equivalent for a fuzzy filter :(i) is fuzzy -fold obstinate.(ii) is fuzzy -fold Boolean and fuzzy prime.
Proof. From Remark 18 and Definitions 17 and 25, if with a proper filter, then is -fold obstinate, hence -fold Boolean and maximal, and thus -fold Boolean and prime. So is fuzzy -fold Boolean and fuzzy prime.
Let such that is a proper filter. Then is prime and -fold Boolean. For each , we have , ; so if and only if .
Remark 30. Since , it follows that any fuzzy -fold Boolean filter is also fuzzy ()-fold Boolean.
5. Fuzzy -Fold Implicative Filter
Definition 31 (see [25]). Let be a residuated lattice.
(i) A filter of is said to be -fold implicative if for all .
In particular a 1-fold implicative filter is an implicative filter.
(ii) is said to be -fold implicative if is an -fold implicative filter.
It is easy to see that is -fold implicative iff, for all , .
Theorem 32. The following conditions are equivalent for a filter of and :(i) is -fold implicative.(ii), .(iii), implies , and implies .
Proof. We give the proof for the binary operation , since the case of is similar.
Note that ; so . Thus, .
, by Proposition 1(12). So from and the definition of a filter, we obtain that implies .
Since , we have .
From Proposition 1((4) and (14)), we have , and taking and we obtain , showing that . Then . Now by Proposition 1(3), we have . By repeating this process, we obtain that .
Corollary 33. (i) Any -fold implicative filter is -fold implicative.
(ii) Any -fold Boolean filter in -fold implicative.
Proof. (i) This clearly appear in the proof of above.
(ii) Suppose is an -fold Boolean filter; then for all . Now, = . Thus .
The case of is similar.
The converse of the above lemma is not true, as illustrated by the following example.
Example 34. Let be the residuated lattice of Example 4:(i)The filter is -fold implicative for all . But is not implicative, since .(ii)The filter is -fold implicative for , but not -fold Boolean, since .
Definition 35. Let be a residuated lattice.
A fuzzy subset is called a fuzzy -fold implicative filter if it satisfies the following conditions:(i)For all , .(ii)For all , .(iii)For all , .
By taking in the definition, we see by Proposition 10 that any fuzzy -fold implicative filter is a fuzzy filter.
Proposition 36. Let be a fuzzy filter. Then, the following conditions are equivalent:(i) is a fuzzy -fold implicative filter.(ii)For all , .(iii)For all , .(iv)For all , and .(v)For all , implies is an -fold implicative filter.
Proof. Let and in condition .
We have = . So . By a similar argument, we have .
Clearly, ; so , showing that by Proposition 11. A similar argument shows that .
Note that ; thus .
If , then ; and similarly, .
If , then . By iteration, we obtain . So .
A similar argument gives .
Let be a filter; then for each , ; and .
Let , and = . Then, and . Now, , and ; thus .
A similar argument shows that .
From Proposition 28, Corollary 33, and Proposition 36, we have the following.
Corollary 37. (i) Any fuzzy -fold implicative filter is fuzzy -fold implicative.
(ii) Any fuzzy -fold Boolean filter is fuzzy -fold implicative.
The converse of this corollary is not true, as it can be seen from Example 38.
Example 38. Let be the residuated lattice of Example 4, a lattice, and in :(i)The fuzzy filter is a fuzzy -fold implicative filter for all , but not a fuzzy implicative filter, since .(ii) is not -fold Boolean, since .
6. Fuzzy -Fold Extended Involutive Filter
Definition 39 (see [25]). Let be a residuated lattice, and .
(i) A filter is an -fold normal filter if it satisfies the following conditions for all : implies , implies .
(ii) is an -fold normal residuated lattice if it satisfies the following conditions: implies , and implies .
Taking in the above definition, we have the following.
Definition 40. (i) A filter is called an -fold extended involutive filter of the residuated lattice if implies and implies , for each .
(ii) is called an -fold extended involutive residuated lattice if, for each , implies and implies .
So, any -fold normal filter is an -fold extended involutive filter.
Proposition 41 (see [25]). Let be a commutative filter of . Then is an -fold normal filter if and only if is an -fold normal residuated lattice.
From the above result, it is easy to see that a residuated lattice is -fold normal (resp., -fold extended involutive) if and only if is an -fold normal (resp., -fold extended involutive) filter of .
Example 42. Let be a lattice such that . Define the operations , , and by the three tables in (5). Then is a residuated lattice which is not a pseudo-BL-algebra, since . Consider and are the only proper filters of . is -fold normal; but is not -fold normal since and .
6.1. Fuzzy -Fold Normal Filter
Definition 43. Let . A fuzzy filter is called a fuzzy -fold normal filter if, for any , implies is an -fold normal filter.
Proposition 44. Let be a fuzzy filter. Then, is a fuzzy -fold normal filter if and only if, for all , and .
Proof. Let be a fuzzy filter.
Assume that is a fuzzy -fold normal filter; let . Then and are -fold normal filters. Now implies , and implies . So, and .
Let such that . If , then implies that .
By a similar argument, we show that implies .
From the above proposition and Proposition 41, we have the following result.
Corollary 45. (i) A commutative fuzzy filter is fuzzy -fold normal if and only if is an -fold normal residuated lattice.
(ii) A filter of is -fold normal if and only if, for all in , is a fuzzy -fold normal filter of .
(iii) is an -fold normal residuated lattice if and only if is a fuzzy -fold normal filter of .
Taking in Proposition 44, we have the following.
Definition 46. Let be a fuzzy filter.
is called a fuzzy -fold extended involutive filter if , for each .
So, any fuzzy -fold normal filter is a fuzzy -fold extended involutive filter.
Example 47. Let be the residuated lattice of Example 42, a lattice, and in . The fuzzy filter is a fuzzy -fold normal filter and is hence fuzzy -fold extended involutive; but the fuzzy filter is not a fuzzy -fold extended involutive filter, since for .
6.2. Fuzzy -Fold Fantastic Filter
Definition 48 (see [25]). Let be a filter of , and .
(i) is -fold fantastic if, for all , .
In particular, any 1-fold fantastic filter is a fantastic filter.
(ii) is called an -fold fantastic residuated lattice if is an -fold fantastic filter of .
It is known that [25] any -fold fantastic filter is -fold normal; and a filter is -fold Boolean iff it is -fold fantastic and -fold implicative.
Taking in the above definition, we have the following.
Definition 49. Let :(i) is an -fold involutive filter of if , for all . In particular, any 1-fold involutive filter is an involutive filter.(ii) is called an -fold involutive residuated lattice if is an -fold involutive filter of .
Example 50. The filter of the residuated lattice of Example 5 is an -fold fantastic filter and is hence -fold involutive.
Definition 51. Let . is a fuzzy -fold fantastic filter if it satisfies the following conditions for all :(i).(ii), and .
In particular, any fuzzy 1-fold fantastic filter is a fuzzy fantastic filter.
If is commutative, a fuzzy fantastic filter is called a fuzzy MV-filter [22].
Example 52. Let be the residuated lattice of Example 5, a lattice, and in . The fuzzy filter is fuzzy -fold fantastic, hence is fuzzy -fold involutive. But the fuzzy filter is not fuzzy -fold fantastic because .
Theorem 53. Let ; the following conditions are equivalent for a fuzzy filter of :(i) is fuzzy -fold fantastic.(ii)For all , and .(iii)Every nonempty -cut of is an -fold fantastic filter.
Proof. Assume that is fuzzy -fold fantastic. Since , we have . By a similar argument, we have .
If , then , showing that . Similarly, we have .
Let ; then ; so is an -fold fantastic filter, and . By Proposition 1(12), ≤ ; so the latter belongs to , and ; this means that .
Similarly, we show that .
From Proposition 1(4) and the fact that , one easily see that any fuzzy -fold fantastic filter is fuzzy -fold fantastic.
Definition 54. Let be a fuzzy filter of .
is called a fuzzy -fold involutive filter if , for all .
Corollary 55. Let be a fuzzy filter of :(i)If is fuzzy -fold fantastic, then it is fuzzy -fold normal and fuzzy -fold involutive.(ii) is fuzzy -fold Boolean if and only if it is fuzzy -fold fantastic and fuzzy -fold implicative.
Proof. This follows from the observation after Definition 48.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors thank the referees, as their comments helped to improve the presentation of the work.