The Scientific World Journal

Volume 2014 (2014), Article ID 929162, 5 pages

http://dx.doi.org/10.1155/2014/929162

Research Article

On Fuzzy Positive Implicative Filters in -Algebras

Sun Shin Ahn¹ and Jeong Soon Han²

¹Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea

²Department of Applied Mathematics, Hanyang University, Ansan 426-791, Republic of Korea

Received 28 March 2014; Accepted 9 May 2014; Published 25 May 2014

Academic Editor: Hee S. Kim

Copyright © 2014 Sun Shin Ahn and Jeong Soon Han. 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.

We study several degrees in defining a fuzzy positive implicative filter, which is a generalization of a fuzzy filter in -algebras.

1. Introduction

In [1], H. S. Kim and Y. H. Kim introduced the notion of a -algebra. Ahn and So [2, 3] introduced the notion of ideals in -algebras. Ahn et al. [4] fuzzified the concept of -algebras and investigated some of their properties. Jun and Ahn [5] provided several degrees in defining a fuzzy implicative filter.

In this paper, we study several degrees in defining a fuzzy positive implicative filter, which is a generalization of a fuzzy filter in -algebras.

2. Preliminaries

We recall some definitions and results discussed in [1–3].

An algebra of type (2, 0) is called a -algebra if

(BE1) for all ,(BE2) for all ,(BE3) for all ,(BE4), for all (exchange).

We introduce a relation “” on a -algebra by , if and only if . A nonempty subset of a -algebra is said to be a subalgebra of , if it is closed under the operation “.” By noticing that , for all , it is clear that . A -algebra is said to be self-distributive, if , for all .

Definition 1 (see [1]). Let be a -algebra and let be a nonempty subset of . Then, is called a filter of if

(F1),(F2) and imply ,

for all .

A nonempty subset of a -algebra is called an implicative filter of if it satisfies (F1) and

(F3) and imply , for all .

Example 2 (see [1]). Let be a -algebra with the following table:

Then is a filter of , but is not a filter of , since and , but .

Proposition 3. Let be a -algebra and let be a filter of . If and , for any , then .

Proposition 4. Let be a self-distributive -algebra. Then, the following hold, for any :

(i)if , then and ;(ii);(iii).

A -algebra is said to be transitive if it satisfies Proposition 4 (iii).

Definition 5 (see [5]). A fuzzy subset of a -algebra is called a fuzzy filter of , if it satisfies, for all ,

(d1),(d2).

A fuzzy subset of a -algebra is called a fuzzy implicative filter of if it satisfies (d1) and

(d3), for all .

Definition 6 (see [5]). Let be a nonempty subset of a -algebra which is not necessarily a filter of . One says that a subset of is an enlarged filter of related to , if it satisfies the following:

(1) is a subset of ,(2),(3).

3. Fuzzy Positive Implicative Filters of -Algebras with Degrees in

denotes a -algebra unless specified otherwise.

Definition 7. A nonempty subset of is called a positive implicative filter of a -algebra if it satisfies (F1) and

(F4) and imply , for all .

Note that every positive implicative filter of a -algebra is a filter of .

Example 8. (1) Let be a self-distributive -algebra ([1]) with the following table:

Then is an implicative filter of but not a positive implicative filter of , since and .

(2) Consider a -algebra as in Example 2. Then is a filter of but not an implicative filter of , since ,, and . Also, it is not a positive implicative filter of , since , and .

(3) Let be a self-distributive -algebra ([5]) with the following table:

Then is an implicative filter of and is a positive implicative filter of .

Definition 9. A fuzzy subset of a -algebra is called a fuzzy positive implicative filter of , if it satisfies (d1) and

(d4), for all .

Definition 10. Let be a nonempty subset of a -algebra which is not necessarily a positive implicative filter of . One says that a subset of is an enlarged positive implicative filter of related to , if it satisfies the following:

(1) is a subset of ,(2),(3).

Obviously, every positive implicative filter is an enlarged positive implicative filter of related to itself. Note that there exists an enlarged positive implicative filter of related to any nonempty subset of .

Example 11. Consider a -algebra which is given in Example 8 (1). Note that is not a positive implicative filter. Then, is an enlarged positive implicative filter of related to and is not a positive implicative filter of , since , and .

Proposition 12. Let be a nonempty subset of a -algebra . Every enlarged positive implicative filter of related to is an enlarged filter of related to .

Proof. Let be an enlarged positive implicative filter of related to . By putting in Definition 10 (3), we have

Hence, is an enlarged filter of related to .

The converse of Proposition 12 is not true in general as seen in the following example.

Example 13. Let be a transitive -algebra ([2]) with the following table:

Let and . Then is an enlarged filter of but it is not an enlarged positive implicative filter of , since and .

In what follows let and be members of , and let and denote a natural number and a real number, respectively, such that , unless otherwise specified.

Definition 14 (see [5]). A fuzzy subset of a -algebra is called a fuzzy filter of with degree , if it satisfies the following:

(e1),(e2).

A fuzzy subset of a -algebra is called a fuzzy implicative filter of with degree , if it satisfies (e1) and

(e3).

Definition 15. A fuzzy subset of a -algebra is called a fuzzy positive implicative filter of with degree , if it satisfies (e1) and

(e4).

Proposition 16 (see [5]). Every fuzzy filter of a -algebra with degree satisfies the following assertions:

(i); (ii); (iii).

Note that if , then a fuzzy positive implicative filter with degree may not be a fuzzy positive implicative filter with degree and vice versa. Obviously, every fuzzy positive implicative filter is a fuzzy positive implicative filter with degree , but the converse may not be true.

Example 17. Consider a self-distributive -algebra which is given in Example 8 (3). Define a fuzzy subset by

Then, is a fuzzy implicative filter of with degree and a fuzzy filter of with degree , but it is neither a fuzzy filter of nor a fuzzy positive implicative filter of with degree since

Example 18. Let be a -algebra ([2]) with the following table:

Define a fuzzy subset by

Then, is a fuzzy positive implicative filter of with degree . But it is neither a fuzzy filter of nor a fuzzy positive implicative filter of with degree since

Also, it is not a fuzzy implicative filter of with degree since

Proposition 19. If is a fuzzy positive implicative filter of a -algebra with degree , then is a fuzzy filter of with degree .

Proof. By putting in (e4), we have

for any . Thus, is a fuzzy filter of with degree .

The converse of Proposition 19 is not true in general (see Example 17).

Note that a fuzzy filter with degree is a fuzzy filter if and only if .

Proposition 20. Let be a fuzzy positive implicative filter of a -algebra with degree . Then, the following holds:

Proof. Assume that is a fuzzy positive implicative filter of a -algebra with degree and let . Using (e4) and (e1), we have

This completes the proof.

Proposition 21. Let be a fuzzy filter of a -algebra with degree satisfying

Then, is a positive implicative filter of with degree .

Proof. Let . Using (e2), we have

Thus, is a positive implicative filter of a -algebra with degree .

Corollary 22. Let be a fuzzy filter of . Then, is a fuzzy positive implicative filter of , if and only if

Proof. It follows from Propositions 20 and 21.

Proposition 23. Every fuzzy positive implicative filter of a -algebra with degree satisfies the following assertions:

(i);(ii).

Proof. It follows from Propositions 16 and 19.

Corollary 24. Let be a fuzzy positive implicative filter of a -algebra with degree . If , then

(i),(ii).

Proposition 25. Let be a self-distributive -algebra . Let be a fuzzy positive implicative filter of with degree . Then,

Proof. Let . Let be a fuzzy positive implicative filter of with degree . By Proposition 19, is a fuzzy filter of with degree . Since , using Proposition 4 (i), we have . Hence,

Using Propositions 16 and 23, we have

This completes the proof.

Definition 26. ([6]) Let be a -algebra. is said to be commutative if the following identity holds:

(C); that is, , where , for all .

Theorem 27. Let be a commutative self-distributive -algebra. Every fuzzy positive implicative filter of with degree is a fuzzy implicative filter of with degree .

Proof. Let be a fuzzy positive implicative filter of with degree . By Proposition 19, is a fuzzy filter of with degree . Using (BE4) and Proposition 4 (iii), we obtain , for any . Hence, by Proposition 16 (iii), we have . On the other hand, using (BE4) and (C), we obtain

Using Proposition 20, we have

This completes the proof

Denote by the set of all positive implicative filters of a -algebra . Note that a fuzzy subset of a -algebra is a fuzzy positive implicative filter of , if and only if

But we know that, for any fuzzy subset of a -algebra , there exist and such that

(1) is a fuzzy positive implicative filter of with degree ,(2).

Example 28. Consider a -algebra which is given in Example 8 (1). Define a fuzzy subset by

If , then is not a positive implicative filter of , since , and . But is a fuzzy positive implicative filter of with degree .

Theorem 29. Let be a fuzzy subset of a -algebra . For any with , if is an enlarged positive implicative filter of related to , then is a fuzzy positive implicative filter of with degree .

Proof. Assume that , for some and . Then and so ; that is, . Since is an enlarged filter of related to , we have ; that is, . This is a contradiction, and thus , for all .

Now suppose that there exist such that . If we take , then . Hence, and . It follows from Definition 10 (3) that so that , which is impossible. Therefore,

for all . Thus, is a fuzzy positive implicative filter of with degree .

Corollary 30. Let be a fuzzy subset of a -algebra . For any with , if is an enlarged positive implicative filter of related to , then is a fuzzy positive implicative filter of with degree .

Theorem 31. Let be such that is not necessarily a positive implicative filter of a -algebra . If is a fuzzy positive implicative filter of with degree , then is an enlarged positive implicative filter of related to .

Proof. Since , we have . Since , there exists and so . By (e1), we obtain . Therefore, .

Let be such that and . Then and . It follows from (e4) that

so that . Thus, is an enlarged positive implicative filter of related to .

Conflict of Interests

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

Acknowledgment

The authors are grateful to the referee for their valuable suggestions and help.

References

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