International Journal of Mathematics and Mathematical Sciences
Volume 2011, Issue 1 754047
Research Article
Open Access

Commutative Pseudo valuations on BCK-Algebras

Myung Im Doh, 

Corresponding Author

Myung Im Doh

Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea gnu.kr

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Min Su Kang, 

Corresponding Author

Min Su Kang

Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea hanyang.ac.kr

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First published: 09 December 2010
https://doi.org/10.1155/2011/754047
Citations: 2
Academic Editor: Young Bae Jun

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 (X; *, 0) of type (2,0) is called a BCI-algebra if it satisfies the following axioms:
  • (i)

    (∀x, y, z ∈ X)  (((x*y)*(x*z))*(z*y) = 0),

  • (ii)

    (∀x, y ∈ X)  ((x*(x*y))*y = 0),

  • (iii)

    (∀x ∈ X)  (x*x = 0),

  • (iv)

    (∀x, y ∈ X)  (x*y = 0,   y*x = 0  ⇒  x = y).

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

    (∀x ∈ X)  (0*x = 0),

then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following conditions:
  • (a1)

    (∀x ∈ X)  (x*0 = x),

  • (a2)

    (∀x, y, z ∈ X)  (x*y = 0⇒(x*z)*(y*z) = 0, (z*y)*(z*x) = 0),

  • (a3)

    (∀x, y, z ∈ X)  ((x*y)*z = (x*z)*y),

  • (a4)

    (∀x, y, z ∈ X)  (((x*z)*(y*z))*(x*y) = 0).

We can define a partial ordering ≤ by x ≤ y if and only if x*y = 0.

A BCK-algebra X is said to be commutative if x∧y = y∧x for all x, y ∈ X where x∧y = y*(y*x).

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

    0 ∈ A,

  • (b2)

    (∀x, y ∈ X)  (x*y ∈ A,   y ∈ A⇒x ∈ A).

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

    (∀x, y, z ∈ X)  ((x*y)*z ∈ A,   z ∈ A⇒x*(y∧x) ∈ A).

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 X denote a BCK-algebra unless otherwise specified.

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

  • (c1)

    (∀x, y ∈ X)(φ(x*y) ≤ φ(x) + φ(y)).

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

  • (c2)

    φ(0) = 0,

  • (c3)

    (∀x, y ∈ X)(φ(x) ≤ φ(x*y) + φ(y)).

Proposition 3.3 (see [4].)For any pseudo valuation φ on X, one has the following assertions:

  • (1)

    φ(x) ≥ 0 for all x ∈ X.

  • (2)

    φ is order preserving,

  • (3)

    φ(x*y) ≤ φ(x*z) + φ(z*y) for all x, y, z ∈ X.

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

  • (c4)

    (∀x, y, z ∈ X)  (φ(x*(y∧x)) ≤ φ((x*y)*z) + φ(z)).

Example 3.5. Let X = {0, a, b, c} be a BCK-algebra with the *-operation given by Table 1. Let ϑ be a real-valued function on X defined by

(3.1)
Routine calculations give that ϑ is a commutative pseudo valuation on X.

Table 1. *-operation.
* 0 a b c
0 0 0 0 0
a a 0 0 a
b b a 0 b
c c c c 0

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

Proof. Let φ be a commutative pseudo valuation on X. For any x, y, z ∈ X, we have

(3.2)
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 X = {0, a, b, c, d} be a BCK-algebra with the *-operation given by Table 2. Let ϑ be a real-valued function on X defined by

(3.3)
Then ϑ is a pseudo valuation on X. Since
(3.4)
ϑ is not a commutative pseudo valuation on X.

Table 2. *-operation.
* 0 a b c d
0 0 0 0 0 0
a a 0 a 0 0
b b b 0 0 0
c c c c 0 0
d d d d c 0

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

Theorem 3.9. For a real-valued function φ on X, the following are equivalent:

  • (1)

    φ is a commutative pseudo valuation on X.

  • (2)

    φ is a pseudo valuation on X that satisfies the following condition:

(3.5)

Proof. Assume that φ is a commutative pseudo valuation on X. Then φ is a pseudo valuation on X by Theorem 3.6. Taking z = 0 in (c4) and using (a1) and (c2) induce the condition (3.5).

Conversely let φ be a pseudo valuation on X satisfying the condition (3.5). Then φ(x*y) ≤ φ((x*y)*z) + φ(z) for all x, y, z ∈ X. It follows from (3.5) that

(3.6)
for all x, y, z ∈ X so that φ is a commutative pseudo valuation on X.

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

(3.7)

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

Proof. Let φ be a pseudo valuation on a commutative BCK-algebra X. Note that

(3.8)
for all x, y, z ∈ X. Hence ((x*(y∧x))*((x*y)*z))*z = 0 for all x, y, z ∈ X. It follows from Lemma 3.10 that φ(x*(y∧x)) ≤ φ((x*y)*z) + φ(z) for all x, y, z ∈ X. Therefore φ is a commutative pseudo valuation on X.

For any real-valued function φ on X, we consider the set
(3.9)

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

Lemma 3.13 (see [7].)For any nonempty subset I of X, the following are equivalent:

  • (1)

    I is a commutative ideal of X.

  • (2)

    I is an ideal of X that satisfies the following condition:

(3.10)

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

Proof. Let φ be a commutative pseudo valuation on a BCK-algebra X. Using Theorem 3.6 and Lemma 3.12, we conclude that Iφ is an ideal of X. Let x, y ∈ X be such that x*y ∈ Iφ. Then φ(x*y) = 0. It follows from (3.5) that φ(x*(y∧x)) ≤ φ(x*y) = 0 so that φ(x*(y∧x)) = 0. Hence x*(y∧x) ∈ Iφ. Therefore Iφ is a commutative ideal of X 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 X = {0, a, b, c} with the *-operation given by Table 3. Let φ be a real-valued function on X defined by

(3.11)
Then Iφ = {0, c} is a commutative ideal of X. Since
(3.12)
φ is not a pseudo valuation on X and so φ is not a commutative pseudo valuation on X.

Table 3. *-operation.
* 0 a b c
0 0 0 0 0
a a 0 0 a
b b a 0 b
c c c c 0

Using an ideal, we establish a pseudo valuation.

Theorem 3.16. For any ideal I of X, we define a real-valued function φI on X by

(3.13)
for all x ∈ X where 0 < t1 < t2. Then φI is a pseudo valuation on X.

Proof. Let x, y ∈ X. If x = 0, then clearly φI(x) ≤ φI(x*y) + φI(y). Assume that x ≠ 0. If y = 0, then φI(x) ≤ φI(x*y) + φI(y). If y ≠ 0, we consider the following four cases:

  • (i)

    x*y ∈ I and y ∈ I,

  • (ii)

    x*y ∉ I and y ∉ I,

  • (iii)

    x*y ∈ I and y ∉ I,

  • (iv)

    x*y ∉ I and y ∈ I.

Case (i) implies that x ∈ I because I is an ideal of X. If x*y = 0, then φI(x*y) = 0 and so φI(x) = t1 = φI(x*y) + φI(y). If x*y ≠ 0, then φI(x*y) = t1 and thus φI(x) = t1 ≤ φI(x*y) + φI(y). The second case implies that φI(x*y) = t2 and φI(y) = t2. Hence φI(x) ≤ t2 < φI(x*y) + φI(y). Let us consider the third case. If x*y = 0, then φI(x*y) = 0 and thus φI(x) ≤ t2 = φI(x*y) + φI(y). If x*y ≠ 0, then φI(x*y) = t1 and so φI(x) ≤ t2 < t1 + t2 = φI(x*y) + φI(y). For the final case, the proof is similar to the third case. Therefore φI is a pseudo valuation on X.

Before ending our discussion, we pose a question.

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

Acknowledgment

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

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