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 .

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 .

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 .

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.