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Abstract and Applied Analysis
Volume 2013, Article ID 736872, 8 pages
http://dx.doi.org/10.1155/2013/736872
Research Article

Some Types of Generalized Fuzzy -Fold Filters in Residuated Lattices

School of Science, Linyi University, Linyi 276005, China

Received 3 November 2013; Accepted 18 December 2013

Academic Editor: Yansheng Liu

Copyright © 2013 Zhen Ming Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Fuzzy filters and their generalized types have been extensively studied in the literature. In this paper, a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences is established, a quotient residuated lattice with respect to generalized fuzzy filter is induced, and several types of generalized fuzzy -fold filters such as generalized fuzzy -fold positive implicative (fantastic and Boolean) filters are introduced; examples and results are provided to demonstrate the relations among these filters.

1. Introduction

Residuated lattices, introduced by Ward and Dilworth in [1], are very basic algebraic structures among algebras associated with logical systems. In fact, many algebras have been proposed as the semantical systems of logical systems, for example, Boolean algebras, MV-algebras, BL-algebras, lattice implication algebras, MTL-algebras, NM-algebras, and -algebras, and so forth, and they are all particular cases of residuated lattices.

Filters are tools of extreme importance in studying these logical algebras and the completeness of nonclassical logics. A filter is also called a deductive system [2]. From logical point of view, various filters correspond to various sets of provable formulae. Filters are also particularly interesting because they are closely related to congruence relations. Hájek [3] introduced the notions of filters and prime filters in BL-algebras and proved the completeness of Basic Logic, BL. Turunen [2] proposed the notions of implicative filters and Boolean filters of BL-algebras and proved that implicative filters are equivalent to Boolean filters in BL-algebras. In [4, 5], some types of filters such as implicative filters, fantastic filters, and Boolean filters in BL-algebras were proposed and some characterizations of them with identity forms were given, and in [6], these filters were generalized to residuated lattices. In [7] some new types of filters such as IMTL-filters, strong MTL-filters, and associative filters were introduced in MTL-algebras, and it was shown in [8] that in any MTL-algebra there exists only one proper associative filter.

At present, there are two branches to generalize the existing types of filters. One is the folding theory and the other is fuzzy sets theory. In the folding approach, in [911], -fold (positive) implicative filters are proposed in BL-algebras. In [12], -fold EIMTL and -fold IMTL-filters of MTL-algebras were defined and some relations between these filters and -fold (positive) implicative filters, -fold fantastic filters, and -fold obstinate filters of MTL-algebras were investigated. In the fuzzy approach, fuzzification ideas have been applied to some fuzzy logical algebras. In [6, 13, 14], fuzzy filters in MTL-algebras, BL-algebras, and residuated lattices were studied, respectively. In particular, several types of fuzzy filters such as fuzzy Boolean filters, fuzzy fantastic filters, fuzzy implicative filters and fuzzy regular filters were introduced in [6, 15]. As a further generalization of some fuzzy notions, in BL-algebras, generalized fuzzy filters [16] which are common generalizations of -fuzzy filters [1719] and -fuzzy filters [16] were investigated, and hemirings were characterized by their -fuzzy ideals [20], -fuzzy ideals [21], and fuzzy ideals with thresholds [22].

This paper continues the study of generalized fuzzy -fold filters in residuated lattices. In Section 2, some basic concepts and properties are recalled, and some new notions about the thresholds are introduced to represent generalized fuzzy filters like fuzzy filters which are convenient to study the properties of generalized fuzzy filters. In Section 3, a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences is established, and a quotient residuated lattice with respect to generalized fuzzy filter is induced. In Section 4, some types of generalized fuzzy -fold filters such as generalized fuzzy -fold positive implicative filters, generalized fuzzy -fold fantastic filters, and generalized fuzzy Boolean filters are introduced, some properties of them are obtained, and examples and results are provided to show the relations among these filters. The last section concludes this paper.

2. Preliminaries

Here, we recall some basic concepts and results which will be used in the sequel. Throughout this paper, will denote a residuated lattice, unless otherwise mentioned.

2.1. Residuated Lattices

Definition 1 (see [1]). An algebra is called a residuated lattice if (1) is a bounded lattice;(2) is a commutative monoid;(3) forms an adjoint pair; that is, if and only if .

In the rest of the paper by we denote .

Lemma 2 (see [3]). Let be a residuated lattice. Then the following properties hold for all :(1);(2);(3);(4) implies and ;(5); (6);(7);(8);(9);(10); in particular, ;(11), where ;(12).

Definition 3 (see [3]). Let be a nonempty subset of a residuated lattice . Then is called a filter if (1) implies ;(2) and imply .

However, a filter can also be alternatively described as for all it holds that and implies [2].

2.2. Generalized Fuzzy Filters

Definition 4 (see [19]). Let be a BL-algebra and such that . Then a fuzzy set of is called a fuzzy filter with thresholds (generalized fuzzy filter for short) if for all it holds that (1);(2) implies .

Definition 5 (see [16]). Let be a generalized fuzzy filter of a BL-algebra . Then is called (1)a generalized fuzzy implicative (or Boolean) filter if for all it holds that (2)a generalized fuzzy positive implicative filter if for all it holds that (3)a generalized fuzzy fantastic filter if for all it holds that

2.3. Properties of the Thresholds α, β and Generalized Fuzzy Filters

In this section, we develop new notions about the thresholds and investigate their properties. Particularly, by these notions, we will verify some properties of generalized fuzzy filters similar to the classical or fuzzy cases.

Here, for all , we denote that ,  , and if and only if and if and only if and .

However, the following lemmas will be useful in the sequel.

Lemma 6. Let such that . Then, for all , implies that(1);(2).

Proof. We only prove (1), (2) can be proven in a similar way. Consider the following: Thus .

Lemma 7. Let such that . Then for all the following assertions are equivalent: (1);(2);(3).

Proof. We only prove (1)(2), (1)(3) can be similarly proved. Assume that . Then it holds that . That is, .
Conversely, assume that . That is, . Obviously, . Thus ; that is, .

Lemma 8. Let such that . Then for all (1);(2).

Proof. Since and , it holds that and . Thus and .

Corollary 9. Let such that . Then for all the following assertions are equivalent: (1);(2);(3).

Proof. It follows immediately from Lemmas 7 and 8.

For and , we have the following transitivity, respectively.

Lemma 10. Let such that . Then for all (1) and imply ;(2) and imply .

Proof. We only prove (1). Assume that and . Using Lemma 6, we have and . Thus . That is, .

Here, the notions of generalized fuzzy filters can be rewritten in residuated lattices as follows.

Definition 11. Let be a residuated lattice and such that . Then a fuzzy set of is called a fuzzy filter with thresholds (generalized fuzzy filter for short) if for all it holds that (1);(2) implies .

Theorem 12. Let be a fuzzy set of . Then is a generalized fuzzy filter if and only if for all it holds that (1);(2).

The items in Definition 11 and Theorem 12 will be frequently used, so we do not cite them every time.

Moreover, the following properties of generalized fuzzy filters will be used in the sequel.

Proposition 13. Let be a generalized fuzzy filter of . Then for all it holds that (1)implies ;(2).

3. Correspondence between Generalized Fuzzy Congruences and Generalized Fuzzy Filters

Here, we define the concept of generalized fuzzy congruence to determine the relationships between generalized fuzzy filters and generalized fuzzy congruences.

Definition 14. Let be a residuated lattice. A fuzzy set of is called a generalized fuzzy congruence of if for all it holds that (1);(2);(3);(4), where .

For a given generalized fuzzy filter , we define as for all .

Obviously, it follows from Proposition 13(2) that .

Proposition 15. Let be a generalized fuzzy filter of . Then is a generalized fuzzy congruence.

Proof. It is trivial.

Let be a generalized fuzzy congruence of and . Define a fuzzy set of as for all which is called a generalized fuzzy congruence class of by . The set is called a generalized fuzzy quotient set by .

Proposition 16. Let be a generalized fuzzy congruence of . Then is a generalized fuzzy filter.

Proof. Consider the following special case of Definition 14(3): . Thus . It follows from Definition 14(3) and (4) that and hence .
Thus is a generalized fuzzy filter.

The following lemma is useful.

Lemma 17. Let be a generalized fuzzy congruence of . Then for all

Proof. It is obvious that .
On the other hand, using Lemma 2(3) and (11), and hence . Similarly, we have . Thus . Then the identity holds.

Proposition 18. Let be a generalized fuzzy congruence of . Then .

Proof. Using Lemma 17, it holds that Thus .

Proposition 19. Let be a generalized fuzzy filter of . Then .

Proof. It is trivial.

Theorem 20. There is a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences.

Proof. It follows immediately from Propositions 18 and 19.

Lemma 21. Let be a generalized fuzzy filter of . Then if and only if for all .

Proof. Assume that for all . Thus . In particular, taking , then .
Conversely, it follows from Lemma 2(11) that and hence . Similarly, we have . Thus ; that is, .

Let be a generalized fuzzy filter of . For any , we define

Theorem 22. Let be a generalized fuzzy filter of . Then is a residuated lattice.

Proof. Routine proofs show that is a residuated lattice.

The above residuated lattice is called a quotient residuated lattice induced by generalized fuzzy filter .

4. Some Types of Generalized Fuzzy -Fold Filters

In this section, we introduce some types of generalized fuzzy -fold filters and investigate the relationships among them.

4.1. Generalized Fuzzy -Fold Positive Implicative Filters

Definition 23. Let be a generalized fuzzy filter of . Then is called a generalized fuzzy -fold positive implicative filter if for all

Example 24. Let . The operations and are defined as Let , . Define as , , , and . It is routine to verify that is a generalized fuzzy 2-fold positive implicative filter.

Theorem 25. Let be a generalized fuzzy filter of . Then the following assertions are equivalent: (1) is a generalized fuzzy -fold positive implicative filter;(2);(3).

Proof. (1)(2) Assume that is a generalized fuzzy -fold positive implicative filter. Taking and , it follows from Lemma 2(2) that . Theorem 12(1) leads to the fact that . Thus .
(2)(3) It is obvious that , and hence . Thus . Obviously, . Thus .
(3)(1) It follows from Lemma 2(8) that . By the same way, we get for all . Using Lemma 2(6), it holds that . Thus . It follows from Lemma 2(5) that , and hence . Thus . It follows from Proposition 13 that . Thus . Thus is a generalized fuzzy -fold positive implicative filter.

Associated with the alternative definitions of generalized fuzzy -fold positive implicative filters, the following properties can be verified.

Example 26. Let . The operations and are defined as Let , . Define as , , , and . Then is not a generalized fuzzy 2-fold positive implicative filter, because .

Proposition 27. Each generalized fuzzy -fold positive implicative filter is a generalized fuzzy -fold positive implicative filer.

Proof. Assume that is a generalized fuzzy -fold positive implicative filter. By Theorem 25(3), it yields that . Using Lemma 2(8), we have , and hence . Thus . It is obvious that . Thus . That is, is a generalized fuzzy -fold positive implicative filter.

It is easy to prove by induction that every fuzzy generalized fuzzy -fold positive implicative filter is a generalized fuzzy -fold positive implicative filter for all integer .

The converse of the above proposition does not always hold.

Example 28. In Example 24, is a generalized fuzzy 2-fold positive implicative filter but not a generalized fuzzy 1-fold positive filter, because .

Proposition 29 (Extension theorem for generalized fuzzy -fold positive implicative filters). Let and be two generalized fuzzy filters of such that and . If is a generalized fuzzy -fold positive implicative filter, then so is .

Proof. Using Theorem 25, we have . By assumption, it holds that . It is obvious that . Thus . Applying Theorem 25 again, is a generalized fuzzy -fold positive implicative filter.

4.2. Generalized Fuzzy -Fold Fantastic Filters

Definition 30. Let be a generalized fuzzy filter of a residuated lattice . Then is called a generalized fuzzy -fold fantastic filter if for all

Example 31. In Example 26, it is routine to verify that is a generalized fuzzy 2-fold fantastic filter.

The following lemma will be useful.

Lemma 32. Let be a residuated lattice. Then for all

Proof. Applying Lemma 2(2) and (10) for times, we get and hence .

Theorem 33. Let be a generalized fuzzy filter of a residuated lattice . Then the following assertions are equivalent: (1) is a generalized fuzzy -fold fantastic filter;(2) for all ;(3) for all .

Proof. (1)(2) Taking , it holds that . Since , we get . Thus .
(2)(3) Replacing with , it follows from Lemma 32 that .
(3)(1) Using Lemma 2(4) and (10), we have , and hence . It is obvious that . Thus . By Proposition 13(1), it yields that . Thus .

Associated with the alternative definitions of generalized fuzzy -fold fantastic filters, the following properties can be verified.

Example 34. In Example 24, is not a generalized fuzzy 2-fold fantastic filter, because .

Proposition 35. Each generalized fuzzy -fold fantastic filter is an -fold generalized fuzzy fantastic filer.

Proof. Assume that is a generalized fuzzy -fold fantastic filter. Using Theorem 33(3), we get . Since , it follows from Lemma 2(4) that , and hence . Thus . It is obvious that . Thus . That is, is an -fold generalized fuzzy fantastic filter.

It is easy to prove by induction that every generalized fuzzy -fold fantastic filter is a generalized fuzzy -fold fantastic filter for all integer .

The converse of the above proposition does not always hold.

Example 36. Let . The operations and are defined as Let , . Define as , , , and . It is routine to verify that is a generalized fuzzy 2-fold fantastic filter, but not a generalized fuzzy 1-fold fantastic filter, because .

Proposition 37 (Extension theorem for generalized fuzzy -fold fantastic filters). Let and be two generalized fuzzy filters of such that and . If is a generalized fuzzy -fold fantastic filter, then so is .

Proof. Using Theorem 33, the proof is similar to that of Proposition 29.

4.3. Generalized Fuzzy -Fold Boolean Filters

Definition 38. Let be a generalized fuzzy filter of . Then is called a generalized fuzzy -fold Boolean filter if for all

Example 39. Let with , , and and let and , and be uncomparable. The operations and are defined as Let , . Define as , , , , , and . It is routine to verify that is a generalized fuzzy 2-fold Boolean filter.

The following lemma will be useful.

Lemma 40. Let be a residuated lattice. Then for all

Proof. By Lemma 2(4), (5), and (8), it holds that and hence .

Theorem 41. Let be a generalized fuzzy filter of . Then the following assertions are equivalent: (1) is a generalized fuzzy -fold Boolean filter;(2) for all ;(3) for all .

Proof. (1)(2) Assume that is a generalized fuzzy -fold Boolean filter. Taking and , we have . Since , it holds that . Thus .
(2)(3) Replacing with , it follows from Lemma 40 that .
(3)(1) Using Lemma 2(5), (3), (6), (4), and (2), we get and hence . It is obvious that . Thus . It follows from Proposition 13 that . Thus .

Associated with the alternative definitions of generalized fuzzy -fold Boolean filters, the following properties can be verified.

Proposition 42. Each generalized fuzzy -fold Boolean filter is an -fold generalized fuzzy Boolean filer.

Proof. Assume that is a generalized fuzzy -fold Boolean filter. It follows from Theorem 41(3) that . Since , it yields that , and hence . It is obvious that . Thus . That is, is an -fold generalized fuzzy Boolean filter.

It is easy to prove by induction that every generalized fuzzy -fold Boolean filter is a generalized fuzzy -fold Boolean filter for all integer .

The converse of the above proposition does not always hold.

Example 43. In Example 39, is a generalized fuzzy 2-fold Boolean filter, but not a generalized fuzzy 1-fold Boolean filter, because .

In order to investigate the relationships among these three types defined above, the following lemma is useful.

Lemma 44. Let be a residuated lattice. Then for all

Proof. Using Lemma 2(1), (2), and (10), we have and hence .

Theorem 45. Let be a generalized fuzzy filter of . Then is a generalized fuzzy -fold Boolean filter if and only if is both a generalized fuzzy -fold fantastic filter and a generalized fuzzy -fold positive implicative filter.

Proof. Assume that is a generalized fuzzy -fold Boolean filter. It follows from Lemma 44 that . It is obvious that . Thus . That is, is a generalized fuzzy -fold positive implicative filter.
Using Lemma 2(2), (3), (4) and (6), we have and hence . Replacing with , we get . By Lemma 2(3), (4), and Lemma 32, it holds that and hence . It is obvious that . Thus . That is, is a generalized fuzzy -fold fantastic filter.
Conversely, it follows from Theorem 25(2) that . Since , it holds that . Using Theorem 33, we get . In particular, taking , it yields that . It follows from Proposition 13 that . Thus . It is obvious that . Thus is a generalized fuzzy -fold Boolean filter.

Obviously, the generalized fuzzy filters in Examples 24 and 26 are not generalized fuzzy 2-fold Boolean filters, because they are not generalized fuzzy 2-fold positive implicative filter and generalized fuzzy 2-fold fantastic filter, respectively.

Proposition 46 (Extension theorem for generalized fuzzy -fold Boolean filters). Let and be two generalized fuzzy filters of such that and . If is a generalized fuzzy -fold Boolean filter, then so is .

Proof. Using Theorem 41, the proof is similar to that of Proposition 29.

5. Conclusions

Generalized fuzzy filters have been extensively studied in the literature. In this paper, we researched the properties of generalized fuzzy filters and introduced some types of generalized fuzzy -fold filters in residuated lattices. We established a correspondence theorem between the set of generalized fuzzy filters and the set of generalized fuzzy congruences and induced a quotient residuated lattice with respect to generalized fuzzy filters. We also defined generalized fuzzy -fold positive implicative, fantastic, and Boolean filters and investigated the correlations among these filters.

In our future work, we will introduce some other types of generalized fuzzy -fold filters in residuated lattices.

Acknowledgment

This research was supported by AMEP of Linyi University, the Natural Science Foundation of Shandong Province (Grant no. ZR2013FL006).

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