`International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 754047, 6 pageshttp://dx.doi.org/10.1155/2011/754047`
Research Article

Commutative Pseudo valuations on BCK-Algebras

1Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea
2Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea

Received 26 September 2010; Accepted 10 November 2010

Copyright © 2011 Myung Im Doh and Min Su Kang. 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 notion of a commutative pseudo valuation on a BCK-algebra is introduced, and its characterizations are investigated. The relationship between a pseudo valuation and a commutative pseudo-valuation is examined.

1. Introduction

D. Buşneag [1] defined pseudo valuation on a Hilbert algebra and proved that every pseudo valuation induces a pseudometric on a Hilbert algebra. Also, D. Buşneag [2] provided several theorems on extensions of pseudo valuations. C. Buşneag [3] introduced the notions of pseudo valuations (valuations) on residuated lattices, and proved some theorems of extension for these (using the model of Hilbert algebras [2]). Using the Buşneag's model, Doh and Kang [4] introduced the notion of a pseudo valuation on BCK/BCI-algebras, and discussed several properties.

In this paper, we introduce the notion of a commutative pseudo valuation on a BCK-algebra, and investigate its characterizations. We discuss the relationship between a pseudo valuation and a commutative pseudo valuation. We provide conditions for a pseudo valuation to be a commutative pseudo valuation.

2. Preliminaries

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

An algebra () of type (2,0) is called a BCI-algebra if it satisfies the following axioms: (i), (ii), (iii), (iv).

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

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

We can define a partial ordering by if and only if .

A BCK-algebra is said to be commutative if for all where .

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

A subset of a BCK-algebra is called a commutative ideal of (see [6]) if it satisfies (b1) and (b3).

We refer the reader to the book in [7] for further information regarding BCK-algebras.

3. Commutative Pseudo Valuations on BCK-Algebras

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

Definition 3.1 (see [4]). A real-valued function on is called a weak pseudo valuation on if it satisfies the following condition: (c1).

Definition 3.2 (see [4]). A real-valued function on is called a pseudo valuation on if it satisfies the following two conditions: (c2), (c3).

Proposition 3.3 (see [4]). For any pseudo valuation on , one has the following assertions: (1) for all . (2) is order preserving, (3) for all .

Definition 3.4. A real-valued function on is called a commutative pseudo valuation on if it satisfies (c2) and (c4).

Example 3.5. Let be a BCK-algebra with the -operation given by Table 1. Let be a real-valued function on defined by Routine calculations give that is a commutative pseudo valuation on .

Table 1: -operation.

Theorem 3.6. In a BCK-algebra, every commutative pseudo valuation is a pseudo valuation.

Proof. Let be a commutative pseudo valuation on . For any , we have This completes the proof.

Combining Theorem 3.6 and [4, Theorem 3.9], we have the following corollary.

Corollary 3.7. In a BCK-algebra, every commutative pseudo valuation is a weak pseudo valuation.

The converse of Theorem 3.6 may not be true as seen in the following example.

Example 3.8. Let be a BCK-algebra with the -operation given by Table 2. Let be a real-valued function on defined by Then is a pseudo valuation on . Since is not a commutative pseudo valuation on .

Table 2: -operation.

We provide conditions for a pseudo valuation to be a commutative pseudo valuation.

Theorem 3.9. For a real-valued function on , the following are equivalent: (1) is a commutative pseudo valuation on . (2) is a pseudo valuation on that satisfies the following condition:

Proof. Assume that is a commutative pseudo valuation on . Then is a pseudo valuation on by Theorem 3.6. Taking in (c4) and using (a1) and (c2) induce the condition (3.5).
Conversely let be a pseudo valuation on satisfying the condition (3.5). Then for all . It follows from (3.5) that for all so that is a commutative pseudo valuation on .

Lemma 3.10 (see [8]). Every pseudo valuation on satisfies the following implication:

Theorem 3.11. In a commutative BCK-algebra, every pseudo valuation is a commutative pseudo valuation.

Proof. Let be a pseudo valuation on a commutative BCK-algebra . Note that for all . Hence for all . It follows from Lemma 3.10 that for all . Therefore is a commutative pseudo valuation on .

For any real-valued function on , we consider the set

Lemma 3.12 (see [4]). If is a pseudo valuation on , then the set is an ideal of .

Lemma 3.13 (see [7]). For any nonempty subset of , the following are equivalent: (1) is a commutative ideal of . (2) is an ideal of that satisfies the following condition:

Theorem 3.14. If is a commutative pseudo valuation on , then the set is a commutative ideal of .

Proof. Let be a commutative pseudo valuation on a BCK-algebra . Using Theorem 3.6 and Lemma 3.12, we conclude that is an ideal of . Let be such that . Then . It follows from (3.5) that so that . Hence . Therefore is a commutative ideal of by Lemma 3.13.

The following example shows that the converse of Theorem 3.14 is not true.

Example 3.15. Consider a BCK-algebra with the -operation given by Table 3. Let be a real-valued function on defined by Then is a commutative ideal of . Since is not a pseudo valuation on and so is not a commutative pseudo valuation on .

Table 3: -operation.

Using an ideal, we establish a pseudo valuation.

Theorem 3.16. For any ideal of , we define a real-valued function on by for all where . Then is a pseudo valuation on .

Proof. Let . If , then clearly . Assume that . If , then . If , we consider the following four cases: (i) and , (ii) and , (iii) and , (iv) and . Case (i) implies that because is an ideal of . If , then and so . If , then and thus . The second case implies that and . Hence . Let us consider the third case. If , then and thus . If , then and so . For the final case, the proof is similar to the third case. Therefore is a pseudo valuation on .

Before ending our discussion, we pose a question.

Question 1. If is commutative ideal of , then is the function in Theorem 3.16 a commutative pseudo valuation on ?

Acknowledgment

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

References

1. D. Buşneag, “Hilbert algebras with valuations,” Mathematica Japonica, vol. 44, no. 2, pp. 285–289, 1996.
2. D. Buşneag, “On extensions of pseudo-valuations on Hilbert algebras,” Discrete Mathematics, vol. 263, no. 1-3, pp. 11–24, 2003.
3. C. Buşneag, “Valuations on residuated lattices,” Annals of the University of Craiova. Mathematics and Computer Science Series, vol. 34, pp. 21–28, 2007.
4. M. I. Doh and M. S. Kang, “BCK/BCI-algebras with pseudo-valuations,” Honam Mathematical Journal, vol. 32, no. 2, pp. 217–226, 2010.
5. Y. S. Huang, BCI-Algebra, Science Press, China, 2006.
6. J. Meng, “Commutative ideals in BCK-algebras,” Pure and Applied Mathematics, vol. 7, no. 2, pp. 49–53, 1991.
7. J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa Co., Seoul, Republic of Korea, 1994.
8. Y. B. Jun, K. J. Lee, and S. S. Ahn, “Positive implicative pseudovaluations on BCK-algebras,” Kybernetika. In press.