Classes of Int-Soft Filters in Residuated Lattices

Jun, Young Bae; Ahn, Sun Shin; Lee, Kyoung Ja

doi:https://doi.org/10.1155/2014/595160

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Research Article | Open Access

Volume 2014 | Article ID 595160 | https://doi.org/10.1155/2014/595160

Classes of Int-Soft Filters in Residuated Lattices

Young Bae Jun,¹Sun Shin Ahn,²and Kyoung Ja Lee³

Academic Editor: Xiaolong Xin

Received12 Jul 2014

Accepted13 Aug 2014

Published27 Aug 2014

Abstract

The notions of int-soft filters, int-soft G-filters, regular int-soft filters, and MV-int-soft filters in residuated lattices are introduced, and their relations, properties, and characterizations are investigated. Conditions for an int-soft filter to be an int-soft G-filter, a regular int-soft filter, or an MV-int-soft filter are provided. The extension property for an int-soft G-filter is discussed. Finally, it is shown that the notion of an MV-int-soft filter coincides with the notion of a regular int-soft filter in BL-algebras.

1. Introduction

In order to deal with fuzzy and uncertain information, nonclassical logic has become a formal and useful tool. As the semantic systems of nonclassical logic systems, various logical algebras have been proposed. Residuated lattices are important algebraic structures which are basic of -algebras, -algebras, -algebras, Gödel algebras, -algebras, lattice implication algebras, and so forth. The (fuzzy) filter theory in the logical algebras has an important role in studying these algebras and completeness of the corresponding nonclassical logics, and it is studied in [1–8]. Uncertainty is an attribute of information. As a new mathematical tool for dealing with uncertainties, Molodtsov [9] introduced the concept of soft sets. Since then several authors studied (fuzzy) algebraic structures based on soft set theory in several algebraic structures. Acar et al. [10] introduced initial concepts of soft rings. Ahn et al. [11] introduced the notion of int-soft filters of a -algebra and investigated related properties. They also discussed characterization of an int-soft filter and solved the problem of classifying int-soft filters by their -inclusive filter. Aktas and Çagman [12] defined soft groups and derived their basic properties using Molodtsov’s definition of the soft sets. Atagün and Sezgin [13] introduced and studied soft subrings and soft ideals of a ring by using Molodtsov’s definition of the soft sets. Moreover, they introduced soft subfields of a field and soft submodule of a left R-module and investigated some related properties about soft substructures of rings, fields, and modules. Çağman and Enginoğlu [14] constructed a uni-int decision making method which selects a set of optimum elements from the alternatives. Feng et al. [15] improved and further extended Çağman and Enginoğlu’s uni-int decision making method in virtue of choice value soft sets and k-satisfaction relations. Çağman and Enginoğlu [16] discussed fuzzy parameterized (FP) soft sets and their related properties and proposed a decision making method based on FP-soft set theory. Feng [17] considered the application of soft rough approximations in multicriteria group decision making problems. Feng et al. [18] initiated the study of soft semirings by using the soft set theory. Jun et al. applied the notion of soft sets by Molodtsov to the theory of -algebras, -algebras, and subtraction algebras (see [19–22]). Jun et al. [23] discussed (strong) intersection-soft filters in -algebras. Zhan and jun [24] investigated characterizations of (implicative, positive implicative, and fantastic) filteristic soft -algebras by means of -soft sets and -soft sets. Recently, Feng and Li [25] explored some relationships among five different types of soft subsets and investigated free soft algebras with respect to soft product operations. They pointed out that soft sets have some nonclassical algebraic properties which are distinct from those of crisp sets or fuzzy sets.

In this paper, we introduce the notions of int-soft filters, int-soft -filters, regular int-soft filters, and -int-soft filters in residuated lattices and investigate their relations and properties. We consider characterizations of int-soft filters, int-soft -filters, regular int-soft filters, and -int-soft filters. We provide conditions for an int-soft filter to be an int-soft -filter, a regular int-soft filter, or an -int-soft filter. We establish the extension property for an int-soft -filter. Finally, we show that the notion of an -int-soft filter coincides with the notion of a regular int-soft filter in -algebras.

2. Preliminaries

Definition 1 (see [1, 26, 27]). A residuated lattice is an algebra of type such that (1) is a bounded lattice;(2) is a commutative monoid;(3) and form an adjoint pair; that is,

In a residuated lattice , the ordering and negation are defined as follows: and for all .

Proposition 2 (see [1, 2, 6, 7, 26, 27]). In a residuated lattice , the following properties are valid:

Definition 3 (see [5]). A nonempty subset of a residuated lattice is called a filter of if it satisfies the conditions

Proposition 4 (see [5]). A nonempty subset of a residuated lattice is a filter of if and only if it satisfies

A soft set theory is introduced by Molodtsov [9], and Çağman and Enginoğlu [14] provided new definitions and various results on soft set theory.

In what follows, let be an initial universe set and let be a set of parameters. Let denote the power set of and .

Definition 5 (see [9, 14]). A soft set over is defined to be the set of ordered pairs where such that if .

3. Int-Soft Filters

In what follows, we take a residuated lattice as a set of parameters.

Definition 6. A soft set over is called an int-soft filter of if it satisfies

Proposition 7. Every int-soft filter of satisfies

Proof. Let . Since , we have by (22). Since , it follows from (23) and (22) that This completes the proof.

Lemma 8. Let be a soft set over that satisfies two conditions (24) and (25). Then one has

Proof. Let be such that . Then , and so Since , (28) is from (27).

Theorem 9. A soft set over is an int-soft filter of if and only if it satisfies two conditions (24) and (25).

Proof. The necessity is from Proposition 7.
Conversely, let be a soft set over that satisfies (24) and (25). If , then and so Since for all , it follows from (28) that for all . Therefore is an int-soft filter of .

Theorem 10. A soft set over is an int-soft filter of if and only if it satisfies condition (27).

Proof. The necessity is from Lemma 8 and Theorem 9.
Conversely, let be a soft set over satisfying (27). Since and for all , it follows from (27) that and for all . Hence is an int-soft filter of by Theorem 9.

Theorem 11. A soft set over is an int-soft filter of if and only if satisfies condition (24) and

Proof. Assume that is an int-soft filter of . Then condition (24) is valid. Using (6) and (25), we have for all .
Conversely, let be a soft set over satisfying (24) and (31). Taking in (31) and using (3), we get for all . Thus is an int-soft filter of by Theorem 9.

Lemma 12. Every int-soft filter over satisfies the following condition:

Proof. If we take and in (25), then This completes the proof.

Theorem 13. A soft set over is an int-soft filter of if and only if it satisfies the following conditions:

Proof. Assume that is an int-soft filter of . Using (3), (24), and (25), we have for all . Using (31) and (34), we get for all .
Conversely, let be a soft set over satisfying two conditions (36) and (37). If we take in (36), then for all . Using (37) induces for all . Therefore is an int-soft filter of by Theorem 9.

Theorem 14. A soft set over is an int-soft filter of if and only if the set is a filter of for all with .

Proof. Assume that is an int-soft filter of . Let and be such that and . Then and . It follows from (24) and (25) that and and so that and . Hence is a filter of by Proposition 4.
Conversely, suppose that is a filter of for all with . For any , let . Then and is a filter of . Hence and so . For any , let and . If we take , then and which imply that . Thus Therefore is an int-soft filter of by Theorem 9.

Theorem 15. For a soft set over , let be a soft set over , where where . If is an int-soft filter of , then so is .

Proof. Suppose that is an int-soft filter of . Then is a filter of for all with by Theorem 14. Thus , and so for all . Let . If and , then . Hence If or , then or . Thus Therefore is an int-soft filter of .

Theorem 16. If is an int-soft filter of , then the set is a filter of for every .

Proof. Since for all , we have . Let be such that and . Then and . Since is an int-soft filter of , it follows from (25) that so that . Hence is a filter of by Proposition 4.

Theorem 17. Let and let be a soft set over . Then (1)if is a filter of , then satisfies the following condition: (2)if satisfies (24) and (48), then is a filter of .

Proof. Assume that is a filter of . Let be such that Then and . Using (20), we have and so .
(2) Suppose that satisfies (24) and (48). From (24) it follows that . Let be such that and . Then and , which imply that . Thus by (48), and so . Therefore is a filter of by Proposition 4.

4. Int-Soft -Filters

Definition 18 (see [28]). A nonempty subset of is called a -filter of if it is a filter of that satisfies the following condition:

Definition 19. A soft set over is called an int-soft -filter of if it is an int-soft filter of that satisfies

Note that condition (51) is equivalent to the following condition:

Lemma 20. Every int-soft filter of satisfies the following condition:

Proof. Let . Using (6) and (8), we have It follows from Theorem 10 that This completes the proof.

Theorem 21. Let be a soft set over . Then is an int-soft -filter of if and only if it is an int-soft filter of that satisfies the following condition:

Proof. Assume that is an int-soft -filter of . Then is an int-soft filter of . Note that , and thus for all . It follows from (22) that . Combining this and (52), we have for all . Using (53) and (57), we have for all .
Conversely, let be an int-soft filter of that satisfies condition (56). If we put and in (56) and use (3) and (24), then for all . Therefore is an int-soft -filter of .

Theorem 22. Let be a soft set over that satisfies condition (24) and Then is an int-soft -filter of .

Proof. If we take in (60) and use (3), then Hence is an int-soft filter of by Theorem 9. Let . Since by (6) and (8), we have by (22). It follows from (22), (24), (25), (8), and (60) that Therefore is an int-soft -filter of by Theorem 21.

The following example shows that any int-soft -filter may not satisfy condition (60).

Example 23. Let (unit interval). For any , define Then is a residuated lattice. Let be a soft set over defined by where . Then is an int-soft -filter of . But it does not satisfy condition (60). For example,

Proposition 24. For an int-soft filter of , condition (60) is equivalent to the following condition:

Proof. Assume that condition (60) is valid. It follows from (24) and (3) that for all .
Conversely, suppose that condition (67) is valid. It follows from (6) and (25) that for all .

Combining Theorem 22 and Proposition 24, we have the following result.

Theorem 25. Every int-soft filter satisfying condition (67) is an int-soft -filter.

Proposition 26. Every int-soft filter of with condition (60) satisfies the following condition:

Proof. Let be an int-soft filter of that satisfies condition (60) and let . Since , that is, , we have by (7). It follows from (8), (6), and (7) that Using (22), (24), (3), (6), and (60), we have Hence condition (70) is valid.

Proposition 27. Every int-soft -filter of with condition (70) satisfies condition (60).

Proof. Let be an int-soft -filter of that satisfies condition (70). For any , we have by (6), (25), (22), (8), (52), and (70). Since , it follows from (22) that and so from (25) that Therefore . Hence condition (60) is valid.

Theorem 28. Let be an int-soft filter of . Then is an int-soft -filter of if and only if the following condition holds:

Proof. Suppose that is an int-soft -filter of . Since for all , we have . It follows from (56) and (3) that and so from (24) that for all .
Conversely, let be an int-soft filter of which satisfies condition (75) and let . Since by (6) and (8), it follows from (22) that Hence, we have by using (25), (75), and (24). Hence is an int-soft -filter of .

Theorem 29. For an int-soft filter of , the following assertions are equivalent: (1) is an int-soft -filter of ;(2).

Proof. . Suppose that is an int-soft -filter of and let . Since , it follows from (22) that . Hence by using (52).
. Assume that holds. Using Lemma 20 and , we have for all , and so is an int-soft -filter of by Theorem 21.

Proposition 30. Every int-soft -filter of satisfies the following conditions:

Proof. Let be an int-soft -filter of . Using (6), (56), (8), and (24), we have for all . Thus (81) holds. Since for all , it follows from (22) that and so that for all by using (81).

Proposition 31. Assume that satisfies the divisibility; that is, , for all . If is an int-soft -filter of satisfying (82), then the following equality is true:

Proof. Using the divisibility and (6), we have for all . It follows from (6) and (82) that for all .

Theorem 32. Let satisfy the divisibility; that is, , for all . Then every int-soft filter of satisfying condition (85) is an int-soft -filter of .

Proof. Using Lemma 20 and (6) and (85), we have for all . Therefore is an int-soft -filter of by Theorem 21.

Theorem 33 (extension property). Let and be int-soft filters of such that ; that is, for all and . If is an int-soft -filter of , then so is .

Proof. Assume that is an int-soft -filter of . Using (6) and (3), we have for all . Thus by hypotheses and (57), and so for all by (24). Since is an int-soft filter of , it follows from (25), (6), and (24) that for all . Therefore is an int-soft -filter of .

5. Regular and -Int-Soft Filters

Zhu and Xu [29] introduced the notion of a regular filter in a residuated lattice.

Definition 34 (see [29]). A filter of is said to be regular if it satisfies the following condition:

Definition 35. An int-soft filter of is said to be regular if it satisfies

Example 36. Let (unit interval). For any , define Then is a residuated lattice (see [29]). Let be a soft set over defined by Then is a regular int-soft filter of .

Theorem 37. For an int-soft filter of , the following assertions are equivalent. (1) is regular.(2). (3).

Proof. Assume that is a regular int-soft filter of and let . Using (7) and (10), we have It follows from (8) and (7) that and so from (24), (94), and (25) that that is, the second condition holds. Since , we have by (8) and (7). It follows from (24), (94), and (25) that Hence the third condition holds. Next, suppose that the second condition is valid. Condition (10) together with the second condition induces for all , and so . Hence is regular. Finally, assume that the third condition is valid. Since for all , it follows from (3) that , and so by (24). Therefore is regular.

Theorem 38. A soft set over is a regular int-soft filter of if and only if it satisfies condition (24) and

Proof. Assume that is a regular int-soft filter of . Clearly condition (24) holds. Using (25) and Theorem 37, we get for all .
Conversely, suppose that satisfies two conditions (24) and (103). Let . Since and , it follows from (3), (24), and (103) that Therefore is an int-soft filter of by Theorem 9. If we take in (103) and use (3) and (24), then Hence is regular by Theorem 37.

By a similar way to the proof of Theorem 38, we have the following characterization of a regular int-soft filter.

Theorem 39. A soft set over is a regular int-soft filter of if and only if it satisfies condition (24) and

Lemma 40 (see [29]). Let be a filter of . Then the following assertions are equivalent: (1) is regular;(2).

Theorem 41. A soft set over is a regular int-soft filter of if and only if the set is a regular filter of for all with .

Proof. Assume that is a regular int-soft filter of . Let be such that . Since is an int-soft filter of , the set is a filter of by Theorem 14. Let be such that . Then by Theorem 37, and so . Hence is regular by Lemma 40.
Conversely, suppose that is a regular filter of for all with . Then is a filter of , and thus is an int-soft filter of by Theorem 14. For any , let . Then which implies from Lemma 40 that . Hence , and so is regular by Theorem 37.

Theorem 42. For any regular filter of , there exist and a regular int-soft filter of such that .

Proof. Let be a soft set over defined by where . Since , we have for all . Let . If and , then by Proposition 4 and Lemma 40. Hence . Suppose that or . Then or , and so . Therefore, by Theorem 38, is a regular int-soft filter of . Obviously, .

Definition 43 (see [29]). A subset of is called an -filter of if it is a filter of that satisfies

Lemma 44 (see [29]). A filter of is an -filter of if and only if it satisfies the condition

Definition 45. A soft set over is called an -int-soft filter of if it is an int-soft filter of with the following additional condition:

Theorem 46. A soft set over is an -int-soft filter of if and only if it satisfies condition (24) and

Proof. Assume that is an -int-soft filter of . Using (25) and (112), we have for all .
Conversely, let be a soft set over which satisfies two conditions (24) and (113). Taking in (113) and using (3) induce condition (25). Hence is an int-soft filter of by Theorem 9. If we take in (113) and use (3) and (24), then we know that satisfies condition (112). Therefore is an -int-soft filter of .

Theorem 47. Let be an int-soft filter of . Then is an -int-soft filter of if and only if the following assertion is valid:

Proof. Assume that is an -int-soft filter of . Then is an int-soft filter of , and so is a filter of for all with by Theorem 14. In particular, is a filter of . Let be such that . Then and so . Therefore is an -filter of , and thus by Lemma 44. Hence , and this and (24) imply that .
Conversely, let be an int-soft filter of that satisfies condition (115). Using (24), (115), (6), and (25), we obtain Therefore is an -int-soft filter of .

Theorem 48. Every -int-soft filter is regular.

Proof. Let be an -int-soft filter of . If we take in (112) and use (3), then and so by (22). Therefore is a regular int-soft filter of .

The converse of Theorem 48 is not true in general as seen in the following example.

Example 49. Let be the residuated lattice which is given in Example 36. Let for any . Note that if then is a regular filter of . But, if then is not an -filter of since , but . Hence the soft set over which is given as follows, is an int-soft filter of which is regular. But, since and , therefore is not an -int-soft filter of .

In a -algebra, that is, a residuated lattice with the following two conditions: the converse of Theorem 48 is true which is shown in the following theorem.

Theorem 50. In a -algebra , the notion of an -int-soft filter coincides with the notion of a regular int-soft filter.

Proof. Based on Theorem 48, it is sufficient to show that every regular int-soft filter is an -int-soft filter. Let be a regular int-soft filter of and let . Then by Theorem 37. Since , we have by (22). Hence It follows that Therefore is an -int-soft filter of .

6. Conclusions

We have introduced the notions of int-soft filters, int-soft -filters, regular int-soft filters, and -int-soft filters in residuated lattices and have investigated their relations and properties. We have considered characterizations of int-soft filters, int-soft -filters, regular int-soft filters, and -int-soft filters. We have provided conditions for an int-soft filter to be an int-soft -filter, a regular int-soft filter, or an -int-soft filter. We have established the extension property for an int-soft -filter and have shown that the notion of an -int-soft filter coincides with the notion of a regular int-soft filter in -algebras. Future research will focus on applying the notions/contents to other algebraic structures.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are deeply grateful to the referees for their valuable comments and suggestions for improving the paper.

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