# Commutative Pseudo valuations on BCK-Algebras

Authors: Myung Im Doh; Min Su Kang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## 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.


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## Body

## 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-algebraX satisfies the following identity:  (v)
(∀x∈X)(0*x=0),thenX 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-algebraX is said to be commutative if x∧y=y∧x for all x,y∈X where x∧y=y*(y*x).A subsetA 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 subsetA 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 letX denote a BCK-algebra unless otherwise specified.Definition 3.1 (see [4]).
A real-valued functionφ 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 functionφ 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 valuationφ 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 functionφ 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.
LetX={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)ϑ=(0abc0799).
Routine calculations give that ϑ is a commutative  pseudo valuation on X.Table 1
*-operation.
*0abc00000aa00abba0bcccc0Theorem 3.6.
In a BCK-algebra, every commutative  pseudo valuation is a pseudo valuation.Proof.
Letφ be a commutative  pseudo valuation on X. For any x,y,z∈X, we have
(3.2)φ(x)=φ(x*(0∧x))≤φ((x*0)*z)+φ(z)=φ(x*z)+φ(z).
This completes the proof.Combining Theorem3.6 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 Theorem3.6 may not be true as seen in the following example.Example 3.8.
LetX={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)ϑ=(0abcd05888).
Then ϑ is a  pseudo valuation on X. Since
(3.4)ϑ(b*(c∧b))=8≰0=ϑ((b*c)*0)+ϑ(0),ϑ is not a commutative  pseudo valuation on X.Table 2
*-operation.
*0abcd000000aa0a00bbb000cccc00ddddc0We provide conditions for a  pseudo valuation to be a commutative pseudo valuation.Theorem 3.9.
For a real-valued functionφ 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)(∀x,y∈X)(φ(x*(y∧x))≤φ(x*y)).Proof.
Assume thatφ 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 letφ 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)φ(x*(y∧x))≤φ(x*y)≤φ((x*y)*z)+φ(z)
for all x,y,z∈X so that φ is a commutative  pseudo valuation on X.Lemma 3.10 (see [8]).
Every  pseudo valuationφ on X satisfies the following implication:
(3.7)(∀x,y,z∈X)((x*y)*z=0⟹φ(x)≤φ(y)+φ(z)).Theorem 3.11.
In a commutative BCK-algebra, every  pseudo valuation is a commutative pseudo valuation.Proof.
Letφ be a  pseudo valuation on a commutative BCK-algebra X. Note that
(3.8)((x*(y∧x))*((x*y)*z))*z=((x*(y∧x))*z)*((x*y)*z)≤(x*(y∧x))*(x*y)=(x∧y)*(y∧x)=0
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 functionφ on X, we consider the set(3.9)Iφ:={x∈X∣φ(x)=0}.Lemma 3.12 (see [4]).
Ifφ is a  pseudo valuation on X, then the set Iφ is an ideal of X.Lemma 3.13 (see [7]).
For any nonempty subsetI 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)(∀x,y∈X)(x*y∈I⟹x*(y∧x)∈I).Theorem 3.14.
Ifφ is a commutative  pseudo valuation on X, then the set Iφ is a commutative ideal of X.Proof.
Letφ 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 Theorem3.14 is not true.Example 3.15.
Consider a BCK-algebraX={0,a,b,c} with the *-operation given by Table 3. Let φ be a real-valued function on X defined by
(3.11)φ=(0abc0370).
Then Iφ={0,c} is a commutative ideal of X. Since
(3.12)φ(b)=7>6=φ(b*a)+φ(a),φ is not a  pseudo valuation on X and so φ is not a commutative  pseudo valuation on X.Table 3
*-operation.
*0abc00000aa00abba0bcccc0Using an ideal, we establish a pseudo valuation.Theorem 3.16.
For any idealI of X, we define a real-valued function φI on X by
(3.13)φI(x)={0ifx=0,t1ifx∈I∖{0},t2ifx∈X∖I
for all x∈X where 0<t1<t2. Then φI is a  pseudo valuation on X.Proof.
Letx,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.
IfI is commutative ideal of X, then is the function φI in Theorem 3.16 a commutative  pseudo valuation on X?



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*Source: 754047-2010-12-09.xml*