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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 616981, 8 pages
http://dx.doi.org/10.1155/2011/616981
Research Article

Graphs Based on BCK/BCI-Algebras

1Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea
2Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea

Received 2 October 2010; Accepted 13 December 2010

Academic Editor: Eun Hwan Roh

Copyright © 2011 Young Bae Jun and Kyoung Ja Lee. 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

The associated graphs of BCK/BCI-algebras will be studied. To do so, the notions of (-prime) quasi-ideals and zero divisors are first introduced and related properties are investigated. The concept of associative graph of a BCK/BCI-algebra is introduced, and several examples are displayed.

1. Introduction

Many authors studied the graph theory in connection with (commutative) semigroups and (commutative and noncommutative) rings as we can refer to references. For example, Beck [1] associated to any commutative ring its zero-divisor graph whose vertices are the zero-divisors of (including 0), with two vertices joined by an edge in case . Also, DeMeyer et al. [2] defined the zero-divisor graph of a commutative semigroup with zero .

In this paper, motivated by these works, we study the associated graphs of BCK/BCI-algebras. We first introduce the notions of (-prime) quasi-ideals and zero divisors and investigated related properties. We introduce the concept of associative graph of a BCK/BCI-algebra and provide several examples. We give conditions for a proper (quasi-) ideal of a BCK/BCI-algebra to be -prime. We show that the associative graph of a BCK-algebra is a connected graph in which every nonzero vertex is adjacent to 0, but the associative graph of a BCI-algebra is not connected by providing an example.

2. Preliminaries

An algebra of type is called a -algebra if it satisfies the following axioms: (I)(II), (III), (IV).

If a BCI-algebra satisfies the following identity: (V),

then is called a -algebra. Any BCK/BCI-algebra satisfies the following conditions: (a1), (a2), (a3), (a4).

We can define a partial ordering ≤ on a BCK/BCI-algebra by if and only if .

A subset of a BCK/BCI-algebra is called an ideal of if it satisfies the following conditions: (b1), (b2).

We refer the reader to the books [3, 4] for further information regarding BCK/BCI-algebras.

3. Associated Graphs

In what follows, let denote a BCK/BCI-algebra unless otherwise specified.

For any subset of , we will use the notations and to denote the sets

Proposition 3.1. Let and be subsets of , then (1) and ,(2)If , then and ,(3) and .

Proof. Let and , then , and so . This says that . Dually, . Hence, (1) is valid.
Assume that and let , then for all , which implies from that for all . Thus, , which shows that . Similarly, we have . Thus, (2) holds.
Using (1) and (2), we have and . If we apply (1) to and , then and . Hence, and .

Definition 3.2. A nonempty subset of is called a quasi-ideal of if it satisfies

Example 3.3. Let be a set with the -operation given by Table 1, then is a BCK-algebra (see [4]). The set is a quasi-ideal of .
Obviously, every quasi-ideal of a BCK-algebra contains the zero element 0. The following example shows that there exists a quasi-ideal of a BCI-algebra such that .

tab1
Table 1: -operation.

Example 3.4. Let be a set with the -operation given by Table 2, then is a BCI-algebra (see [3]). The set is a quasi-ideal of containing the zero element 0, but the set is a quasi-ideal of which does not contain the zero element 0.
Obviously, every ideal of is a quasi-ideal of , but the converse is not true. In fact, the quasi-ideal in Example 3.3 is not an ideal of . Also, quasi-ideals and in Example 3.4 are not ideals of .

tab2
Table 2: -operation.

Definition 3.5. A (quasi-) ideal of is said to be -prime if it satisfies (i) is proper, that is, ,(ii).

Example 3.6. Consider the BCK-algebra with the operation which is given by the Table 3, then the set is an -prime ideal of .

tab3
Table 3: -operation.

Theorem 3.7. A proper (quasi-) ideal of is -prime if and only if it satisfies for all .

Proof. Assume that is an -prime (quasi-) ideal of . We proceed by induction on . If , then the result is true. Suppose that the statement holds for . Let be such that . If , then . Assume that , then by the -primeness of , which shows that . Using the induction hypothesis, we conclude that for some . The converse is clear.

For any , we will use the notation to denote the set of all elements such that , that is, which is called the set of zero divisors of .

Lemma 3.8. If is a BCK-algebra, then for all .

Proof. Let and , then , and so for all .

If is a BCI-algebra, then Lemma 3.8 does not necessarily hold. In fact, let be a set with the -operation given by Table 4, then is a BCI-algebra (see [4]). Note that for all and for all .

tab4
Table 4: -operation.

Corollary 3.9. If is a BCI-algebra, then for all with .

Lemma 3.10. If is a BCI-algebra, then for all , where is the BCK-part of .

Proof. Straightforward.

Lemma 3.11. For any elements and of a BCK-algebra , if , then and .

Proof. Assume that . Let , then , and so Thus, , which shows that . Obviously, .

Theorem 3.12. For any element of a BCK-algebra , the set of zero divisors of is a quasi-ideal of containing the zero element 0. Moreover, if is maximal in , then is -prime.

Proof. By Lemma 3.8, we have . Let and be such that . Using Lemma 3.11, we have and so . Hence, . Therefore, is a quasi-ideal of . Let be such that and , then . Let be an arbitrarily element, then , and so , that is, . Since , we have . It follows from Lemma 3.11 that so from the maximality of it follows that . Hence, , which shows that is -prime.

Definition 3.13. By the associated graph of a BCK/BCI-algebra , denoted , we mean the graph whose vertices are just the elements of , and for distinct , there is an edge connecting and , denoted by if and only if .

Example 3.14. Let be a set with the -operation given by Table 5, then is a BCK-algebra (see [4]). The associated graph of is given by the Figure 1.

tab5
Table 5: -operation.
616981.fig.001
Figure 1: Associated graph of .

Example 3.15. Let be a set with the -operation given by Table 6, then is a BCK-algebra (see [4]). By Lemma 3.8, each nonzero point is adjacent to 0. Note that , , and . Hence the associated graph of is given by the Figure 2.

tab6
Table 6: -operation.
616981.fig.002
Figure 2: Associated graph of .

Example 3.16. Let be a set with the -operation given by Table 7, then is a BCK-algebra (see [4]). By Lemma 3.8, each nonzero point is adjacent to 0. Note that , that is, 1 is not adjacent to 2 and . Hence, the associated graph of is given by Figure 3.

tab7
Table 7: -operation.
616981.fig.003
Figure 3: Associated graph of .

Example 3.17. Consider a BCI-algebra with the -operation given by Table 4, then , and . Since , we know from Lemma 3.10 that two points 1 and 2 are adjacent to 0. The associated graph of is given by Figure 4.

616981.fig.004
Figure 4: Associated graph of .

Theorem 3.18. Let be the associated graph of a BCK-algebra . For any , if and are distinct -prime quasi-ideals of , then there is an edge connecting and .

Proof. It is sufficient to show that . If , then and . For any , we have . Since is -prime, it follows that so that . Similarly, . Hence, , which is a contradiction. Therefore, is adjacent to .

Theorem 3.19. The associated graph of a BCK-algebra is connected in which every nonzero vertex is adjacent to 0.

Proof. It follows from Lemma 3.8.

Example 3.17 shows that the associated graph of a proper BCI-algebra may not be connected.

4. Conclusions

We have introduced the associative graph of a BCK/BCI-algebra with several examples. We have shown that the associative graph of a BCK-algebra is connected, but the associative graph of a BCI-algebra is not connected.

Our future work is to study how to induce BCK/BCI-algebras from the given graph (with some additional conditions).

References

  1. I. Beck, “Coloring of commutative rings,” Journal of Algebra, vol. 116, no. 1, pp. 208–226, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. R. DeMeyer, T. McKenzie, and K. Schneider, “The zero-divisor graph of a commutative semigroup,” Semigroup Forum, vol. 65, no. 2, pp. 206–214, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Y. S. Huang, BCI-Algebra, Science press, Beijing, China, 2006.
  4. J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa Co., Seoul, Korea, 1994.