International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 628931, 8 pages

doi:10.1155/2012/628931

Research Article

Fuzzy -Fold BCI-Positive Implicative Ideals in BCI-Algebras

S. F. Tebu,¹ C. Lele,² and J. B. Nganou³

¹Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon

²Department of Mathematics, University of Dschang, P.O. Box 67, Dschang, Cameroon

³Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

Received 11 June 2012; Accepted 21 August 2012

Academic Editor: Siegfried Gottwald

Copyright © 2012 S. F. Tebu et al. 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 consider the notion of fuzzy -fold positive implicative ideals in BCI-algebras. We analyse many properties of fuzzy -fold positive implicative ideals. We also establish an extension properties for fuzzy n-fold positive implicative ideals in BCI-algberas. This work generalizes the corresponding results in the crisp case.

1. Introduction

Liu and Zhang [1], Wei and Jun [2] independently introduced BCI-positive implicative ideals and used these to completely describe positive implicative BCI-algebras (namely, weakly positive implicative BCI-algebras). Tebu et al. [3] introduced the notion of -fold positive implicative ideals in BCI-algebra. Fuzzy ideals are useful tool to obtain results on BCI-algebras. The sets of provable formulas in the corresponding inference systems from the point of view of uncertain information can be described by fuzzy ideals of those algebraic semantics. So in this paper, we discuss the notion of fuzzy -fold positive implicative ideals in BCI-algebras. We show that every fuzzy -fold positive implicative ideal is a fuzzy ideal, and give a condition for a fuzzy ideal to be a fuzzy -fold positive implicative ideal. Using the level set, we provide a characterization of fuzzy -fold positive implicative ideals. Finally, we establish an extension property for fuzzy -fold positive implicative ideal. This paper generalizes the corresponding results in BCK/BCI-algebras [4].

2. Preliminaries

We recollect some definitions and results which will be used in the following, and we shall not cite them every time they are used.

Definition 2.1 (see [5]). An algebra of type , is said to be a BCI-algebra if it satisfies the following conditions for all :

(i)BCI1-;(ii)BCI2-;(iii)BCI3-;(iv)BCI4- and imply .

If a BCI-algebra X satisfies the condition for all ; then X is called a BCK-algebra. Hence, BCK-algebra form a subclass of BCI-algebra.

Proposition 2.2 (see [5]). On every BCI-algebra , there is a natural order called the BCI-ordering defined by if and only if . Under this order, the following axioms hold for all .

(a); (b)if , then and ;(c); (d); (e); (f); (g); (h); (i); (j)if , then ;(k)if , then .

Let be a positive integer. Throughout this paper we appoint that denotes a BCI-algebra; , in which occurs times; and denotes , where .

Definition 2.3 (see [5]). A nonempty subset of a BCI-algebra is called an ideal of if it satisfies:

() ,

() and imply .

Definition 2.4 (see [5]). A nonempty subset of a BCI-algebra is called a BCI-positive implicative ideal of if it satisfies and

() and imply ; for all , .

We recall [6, 7] that a fuzzy set of a set is a function . For a fuzzy set in and , define to be the set

Definition 2.5 (see [8]). (a) Let be a BCI-algebra. A fuzzy set in is said to be a fuzzy ideal of if:

(i) for all ;

(ii) ; for all .

Note that every ideal is order reversing, that is, if then .

(b) A fuzzy set in is called a fuzzy positive implicative ideal of if it satisfies:

(i) for all .(ii); for all .

Definition 2.6. Let be a BCK-algebra and be a positive integer. Then the fuzzy set of is called a fuzzy -fold positive implicative ideal in BCK-algebra if it satisfies the following conditions:

(i), for all ;(ii); for all .

Definition 2.7 (see [9]). Let be a BCK-algebra and be a positive integer. Then is called an -fold positive implicative BCK-algebra if ; for all .

Definition 2.8 (see [3]). Let be a BCI-algebra and be a positive integer. Then is called an -fold positive implicative BCI-algebra if ; for all .

Definition 2.9 (see [3]). A nonempty subset of a BCI-algebra is called an -fold BCI-positive implicative ideal of if it satisfies and

and imply ; for all , .

3. Fuzzy -Fold Positive Implicative Ideals in BCI-Algebras

In this section, we introduce the notion of fuzzy -fold positive implicative ideal in a BCI-algebra and study some important properties.

Definition 3.1. A fuzzy set of a BCI-algebra is called a fuzzy -fold positive implicative ideal of if it satisfies the following conditions:

(i) for all ;(ii); for all , .

Remark 3.2. (a) Notice that fuzzy -fold positive implicative ideal in BCI-algebra is a fuzzy positive implicative ideal in BCI-algebra.

(b) The notion of fuzzy -fold positive implicative ideal in a BCI-algebras generalizes the notion of fuzzy -fold positive implicative ideal in BCK-algebras. This is because if is a BCK-algebra, for every , .

(c) Every fuzzy -fold positive implicative ideal in BCI-algebra is a fuzzy ideal.

The following example shows that the converse of Remark 3.2 (c) may not be true.

Example 3.3. Consider a BCI-algebra with Cayley table as follows:

We define in given by , . is a fuzzy ideal, but not a fuzzy -fold positive implicative ideal of . As , but .

Remark 3.4. In an -fold positive implicative BCI-algebra, every fuzzy ideal is a fuzzy -fold positive implicative.

Theorem 3.5. Let be a fuzzy ideal of a BCI-algebra . Then the following conditions are equivalent:

(i), for all , ;(ii) is a fuzzy -fold positive implicative ideal;(iii), for all .

Proof. (i)⇒(ii). Let . We will prove that

Since is a fuzzy ideal, we have ; .

(ii)⇒(iii). It is easy to obtain by setting .

(iii)⇒(i). Let with and , it suffices to show that .

Thus, , it follows ; . Hence, .

Theorem 3.6. Let be a BCI-algebra and be a fuzzy set in .

If is a fuzzy -fold positive implicative ideal of , then .

Proof. Suppose that is a fuzzy -fold positive implicative ideal of . putting , then , for all .

We have , since is a fuzzy -fold positive implicative ideal, we obtain . The proof is complete.

However the following example proves that the converse of Theorem 3.6 is not true.

Example 3.7. Consider a BCI-algebra with the following Cayley table:

We define : in by and for all   μ . It is clear that for all   ; but is not a fuzzy -fold positive implicative ideal because , but .

Lemma 3.8. Let be a fuzzy ideal in . If is a fuzzy -fold positive implicative ideal, then is a fuzzy -fold positive implicative ideal.

Proof. let be a fuzzy -fold positive implicative ideal of , let . Then . Therefore, . Hence, is a fuzzy -fold positive implicative ideal.

Using this lemma and a simple induction argument, we obtain the following proposition.

Proposition 3.9. Let be a fuzzy ideal of a BCI-algebra and . If is a fuzzy -fold positive implicative ideal, then is a fuzzy -fold positive implicative ideal.

However there exists fuzzy -fold positive implicative ideals which are not fuzzy -fold positive implicative ideals.

Example 3.10. Consider the set , and define the operation on by:

Consider the fuzzy ideal define by and for all , . Then is a fuzzy -fold positive implicative ideal, but not a fuzzy positive implicative ideal of because and .

Now, we established some transfer principle for fuzzy -fold positive implicative ideals in BCI-algebras.

Proposition 3.11. Let be a fuzzy ideal and be an ideal of a BCI-algebra .

Then is an -fold positive implicative ideal if and only if it‘s characteristic function is a fuzzy -fold positive implicative ideal.

Proof. Assume that is an -fold positive implicative ideal. Let . If , then . If , then because is an -fold positive implicative ideal. So, we also have .

Conversely, suppose that is a fuzzy -fold positive implicative ideal. Let such that . Then ; since is a fuzzy -fold positive implicative ideal, . We conclude that .

Theorem 3.12. Let be a fuzzy ideal in a BCI-algebra . Then is a fuzzy -fold positive implicative if and only if the -cut set either is empty or is an -fold positive implicative ideal.

Proof. Suppose that is a fuzzy -fold positive implicative ideal of and , for all . We will show that is an -fold positive implicative ideal. Since is nonempty, there exists such that ; since , for all , we have: ; thus .

Let ; such that . Then , since is fuzzy -fold positive implicative ideal, we have . Hence and is an -fold BCI-positive implicative ideal.

Conversely, assume that for all and is an -fold BCI-positive implicative ideal. We will show that is a fuzzy -fold positive implicative ideal.

It is easy to see that , for all .

Now assume that there exist such that .

setting then ; it follows that , but . This is a contradiction. Hence is a fuzzy -fold positive implicative ideal.

Theorem 3.13. If is an -fold fuzzy positive implicative ideal of a BCI-algebra , then the set is an -fold positive implicative ideal of .

Proof. Let be an -fold fuzzy positive implicative ideal of . Clearly .

Let such that . Then . It follows by for all , that so that . Hence is an -fold positive implicative ideal of .

Theorem 3.14 (extension property for fuzzy -fold positive implicative ideals in BCI-algebra). Let , two fuzzy ideals of a BCI-algebra such that and , that is, for all . If is a fuzzy -fold positive implicative ideal then is also fuzzy -fold positive implicative ideal.

Proof. Using Theorem 3.5, it is sufficient to show that satisfies the inequality for all .

Let ; setting , we have

Since is a fuzzy ideal, .

Hence by Theorem 3.5, is a fuzzy -fold positive implicative ideal of .

Definition 3.15 (see [10]). Let be two BCI-algebras.

A map is called a BCI-homomorphism if: for all .

Definition 3.16. Let and be two BCI-algebras, a fuzzy subset of , a fuzzy subset of and a BCI-homomorphism.

The image of under denoted by is a fuzzy set of defined by: For all , if and if .

The preimage of under f denoted by is a fuzzy set of defined by: For all ,. Let be an onto BCI-homomorphism, It is easy to prove that the preimage of a fuzzy -fold positive implicative ideal under is also a fuzzy -fold positive implicative ideal.

Definition 3.17. A fuzzy subset of has a sup property if for any nonempty subset of , there exists such that . Using this fact, we can prove the following result:

Proposition 3.18. Let be an onto BCI-homomorphism, the image of a fuzzy -fold positive implicative ideal with a sup property is also a fuzzy -fold positive implicative ideal.

4. Conclusion

To investigate the structure of an algebraic system, it is clear that ideals with special properties play an important role. The present paper introduced and studied the notion of fuzzy -fold positive implicative ideal in BCI-algebra. The extension property of fuzzy -fold positive implicative ideals was established. The main purpose of future work is to investigate the fuzzy foldness of other types of ideals in BCI-algebras and the relation diagram between them similar to the one in [8, 11, 12]. We are also doing some investigations on logic whose algebraic semantics is -fold positive implicative BCI-algebras.

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