International Journal of Mathematics and Mathematical Sciences

Volume 2010, Article ID 819463, 11 pages

http://dx.doi.org/10.1155/2010/819463

## Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the Theory of Falling Shadows

^{1}Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea^{2}Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea^{3}Department of Mathematics, University of Ulsan, Ulsan 680-749, Republic of Korea

Received 11 March 2010; Revised 14 November 2010; Accepted 20 December 2010

Academic Editor: Heinz Gumm

Copyright © 2010 Young Bae Jun 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

Based on the theory of falling shadows and fuzzy sets, the notion of a falling fuzzy implicative ideal of a BCK-algebra is introduced. Relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals are given. Relations between fuzzy implicative ideals and falling fuzzy implicative ideals are provided.

#### 1. Introduction and Preliminaries

##### 1.1. Introduction

In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [1] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [2] introduced the theory of falling shadows which directly relates probability concepts with the membership function of fuzzy sets. Falling shadow representation theory shows us the way of selection relaid on the joint degrees distributions. It is reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. The mathematical structure of the theory of falling shadows is formulated in [3]. Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Jun and Park [6] discussed the notion of a falling fuzzy subalgebra/ideal of a BCK/BCI-algebra. Jun and Kang [7, 8] also considered falling fuzzy positive implicative ideals and falling fuzzy commutative ideals. In this paper, we establish a theoretical approach to define a fuzzy implicative ideal in a BCK-algebra based on the theory of falling shadows. We consider relations between fuzzy implicative ideals and falling fuzzy implicative ideals. We provide relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals.

##### 1.2. Basic Results on BCK-Algebras and Fuzzy Aspects

A BCK/BCI-algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers.

An algebra of type is called a *BCI-algebra* if it satisfies the following conditions: (i), (ii), (iii), (iv). If a BCI-algebra satisfies the following identity: (v),
then is called a *-algebra*. Any BCK-algebra satisfies the following axioms: (a1),
(a2),
(a3),
where if and only if .A subset of a BCK-algebra is called an *ideal* of if it satisfies the following: (b1),
(b2). Every ideal of a BCK-algebra has the following assertion:
A subset of a BCK-algebra is called a *positive implicative ideal* of if it satisfies (b1) and (b3). A subset of a BCK-algebra is called a *commutative ideal* of if it satisfies (b1) and (b4). A subset of a BCK-algebra is called an *implicative ideal* of if it satisfies (b1) and (b5). We refer the reader to the paper [9] and book [10] for further information regarding BCK-algebras.A fuzzy set in a BCK-algebra is called a *fuzzy ideal* of (see [11]) if it satisfies the following: (c1),
(c2). A fuzzy set in a BCK-algebra is called a *fuzzy positive implicative ideal* of (see [12]) if it satisfies (c1) and (c3). A fuzzy set in a BCK-algebra is called a *fuzzy commutative ideal* of (see [13]) if it satisfies (c1) and (c4). A fuzzy set in a BCK-algebra is called a *fuzzy implicative ideal* of (see [14]) if it satisfies (c1) and (c5).

Proposition 1.1 (see [11, 14]). *Let be a fuzzy set in a BCK-algebra . Then is a fuzzy (implicative) ideal of if and only if
**
where .*

##### 1.3. The Theory of Falling Shadows

We first display the basic theory on falling shadows. We refer the reader to the papers [1–5] for further information regarding falling shadows.

Given a universe of discourse , let denote the power set of . For each , let
and for each , let
An ordered pair is said to be a *hyper-measurable structure* on if is a -field in and . Given a probability space and a hyper-measurable structure on , a *random set* on is defined to be a mapping which is - measurable, that is,
Suppose that is a random set on . Let
Then is a kind of fuzzy set in . We call a *falling shadow* of the random set , and is called a *cloud* of .

For example, , where is a Borel field on and is the usual Lebesgue measure. Let be a fuzzy set in and be a -cut of . Then
is a random set and is a cloud of . We will call defined above as the *cut-cloud* of (see [1]).

#### 2. Falling Fuzzy Implicative Ideals

In what follows let denote a BCK-algebra unless otherwise.

*Definition 2.1 (see [6–8]). *Let be a probability space, and let
be a random set. If is an ideal (resp., positive implicative ideal and commutative ideal) of for any , then the falling shadow of the random set , that is,
is called a *falling fuzzy ideal* (resp., *falling fuzzy positive implicative ideal* and *falling fuzzy commutative ideal*) of .

Let be a probability space and let
where is a BCK-algebra. Define an operation on by
for all . Let be defined by for all . Then is a BCK-algebra (see [6]).

*Definition 2.2. *Let be a probability space and let
be a random set. If is an implicative ideal of for any , then the falling shadow of the random set , that is,
is called a *falling fuzzy implicative ideal* of .

For any subset of and , let
Then .

Theorem 2.3. *If is an implicative ideal of , then
**
is an implicative ideal of .*

*Proof. *Assume that is an implicative ideal of and let . Since , we see that . Let be such that and . Then
and . Since is an implicative ideal of , it follows from (b5) that and so . Hence is an implicative ideal of .

Since we see that is a random set on . Let Then is a falling fuzzy implicative ideal of .

*Example 2.4. *Consider a BCK-algebra with a Cayley table which is given by Table 1 (see [10, page 274]). Let and let be defined by
Then is an implicative ideal of for all . Hence , which is given by , is a falling fuzzy implicative ideal of , and it is represented as follows:
Then
If , then is not an implicative ideal of since and , but . It follows from Proposition 1.1 that is not a fuzzy implicative ideal of .

Theorem 2.5. *Every fuzzy implicative ideal of is a falling fuzzy implicative ideal of .*

*Proof. *Consider the probability space , where is a Borel field on and is the usual Lebesque measure. Let be a fuzzy implicative ideal of . Then is an implicative ideal of for all by Proposition 1.1. Let
be a random set and for every . Then is a falling fuzzy implicative ideal of .

Example 2.4 shows that the converse of Theorem 2.5 is not valid.

Theorem 2.6. *Every falling fuzzy implicative ideal is a falling fuzzy ideal.*

*Proof. *Let be a falling fuzzy implicative ideal of . Then is an implicative ideal of , and hence it is an ideal of . Thus is a falling fuzzy ideal of .

The converse of Theorem 2.6 is not true in general as shown by the following example.

*Example 2.7. *Consider a BCK-algebra with a Cayley table which is given by Table 2 (see [10, page 260]). Let and let be defined by
Then is an ideal of for all . Hence is a falling fuzzy ideal of , and
In this case, we can easily check that is a fuzzy ideal of (see [6]). If , then is not an implicative ideal of since and , but . Therefore is not a falling fuzzy implicative ideal of .

Theorem 2.8. *Every falling fuzzy implicative ideal is both a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal.*

*Proof. *Let be a falling fuzzy implicative ideal of . Then is an implicative ideal of , and hence it is both a positive implicative ideal and a commutative ideal of . Thus is both a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal.

The following example shows that the converse of Theorem 2.8 may not be true.

*Example 2.9. *(1) Consider a BCK-algebra with a Cayley table which is given by Table 3 (see [10, page 263]). Let and let be defined by
Then is a commutative ideal of for all . Hence , which is given by , is a falling fuzzy commutative ideal of (see [8]). But it is not a falling fuzzy implicative ideal of because if then is not an implicative ideal of since and , but .

(2) Let be a BCK-algebra in which the -multiplication is defined by Table 4. Let and let
Then is a positive implicative ideal of for all . Thus , which is given by , is a falling fuzzy positive implicative ideal of . But it is not a falling fuzzy implicative ideal of because if then is not an implicative ideal of since and , but .

The notions of a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal are independent, that is, a falling fuzzy commutative ideal need not be a falling fuzzy positive implicative ideal, and vice versa. In fact, the falling fuzzy commutative ideal in Example 2.9(1) is not a falling fuzzy positive implicative ideal. Also the falling fuzzy positive implicative ideal in Example 2.9(2) is not a falling fuzzy commutative ideal.

Let be a probability space and a falling shadow of a random set . For any , let
Then .

Proposition 2.10. *If a falling shadow of a random set is a falling fuzzy implicative ideal of , then *(1)*,
*(2)*. *

*Proof. *(1) Let . Then and . Since is an implicative ideal of , it follows from (b5) that so that . Therefore
for all .

(2) If , then . Since
we have . Since is an implicative ideal and hence an ideal of , it follows that and so . Hence for all .

Theorem 2.11. *If is a falling fuzzy implicative ideal of , then
**
where for any .*

*Proof. *By Definition 2.2, is an implicative ideal of for any . Hence
and thus
Therefore
This completes the proof.

Theorem 2.11 means that every falling fuzzy implicative ideal of is a -fuzzy implicative ideal of .

#### Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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