Next Article in Journal
Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator
Previous Article in Journal
On the Analysis of Mixed-Index Time Fractional Differential Equation Systems
Article Menu

Export Article

Axioms 2018, 7(2), 26; doi:10.3390/axioms7020026

Article
Quotient Structures of BCK/BCI-Algebras Induced by Quasi-Valuation Maps
1
Department of Mathematics, Jeju National University, Jeju 63243, Korea
2
Department of Mathematical Sciences, Shahid Beheshti University, Tehran 1983969411, Iran
3
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
*
Author to whom correspondence should be addressed.
Received: 4 February 2018 / Accepted: 11 April 2018 / Published: 23 April 2018

Abstract

:
Relations between I-quasi-valuation maps and ideals in B C K / B C I -algebras are investigated. Using the notion of an I-quasi-valuation map of a B C K / B C I -algebra, the quasi-metric space is induced, and several properties are investigated. Relations between the I-quasi-valuation map and the I-valuation map are considered, and conditions for an I-quasi-valuation map to be an I-valuation map are provided. A congruence relation is introduced by using the I-valuation map, and then the quotient structures are established and related properties are investigated. Isomorphic quotient B C K / B C I -algebras are discussed.
Keywords:
ideal; I-quasi-valuation map; I-valuation map; quasi-metric
MSC:
06F35; 03G25; 03C05

1. Introduction

B C K / B C I -algebras are an important class of logical algebras introduced by Imai and Iséki (see [1,2,3,4]), and have been extensively investigated by several researchers. It is known that the class of B C K -algebras is a proper subclass of B C I -algebras. Song et al. [5] introduced the notion of quasi-valuation maps based on a subalgebra and an ideal in B C K / B C I -algebras, and then they investigated several properties. They provided relations between a quasi-valuation map based on a subalgebra and a quasi-valuation map based on an ideal, and gave a condition for a quasi-valuation map based on an ideal to be a quasi-valuation map based on a subalgebra in B C I -algebras. Using the notion of a quasi-valuation map based on an ideal, they constructed (pseudo) metric spaces, and showed that the binary operation ∗ in B C K -algebras is uniformly continuous.
In this paper, we discuss relations between I-quasi-valuation maps and ideals in B C K / B C I -algebras. Using the notion of an I-quasi-valuation map of a B C K / B C I -algebra, we induce the quasi-metric space, and investigate several properties. We discuss relations between the I-quasi-valuation map and the I-valuation map. We provide conditions for an I-quasi-valuation map to be an I-valuation map. We use I-quasi-valuation maps to introduce a congruence relation, and then we construct the quotient structures and investigate related properties. We establish isomorphic quotient B C K / B C I -algebras.

2. Preliminaries

By a B C I -algebra, we mean a nonempty set X with a binary operation ∗ and a special element 0 satisfying 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 B C K -algebra. Any B C K / B C I -algebra X satisfies the following conditions:
( x X ) ( x 0 = x ) ,
( x , y , z X ) ( x y = 0 ( x z ) ( y z ) = 0 , ( z y ) ( z x ) = 0 ) ,
( x , y , z X ) ( ( x y ) z = ( x z ) y ) ,
( x , y , z X ) ( ( ( x z ) ( y z ) ) ( x y ) = 0 ) .
Any B C I -algebra X satisfies the following condition:
( x , y X ) ( 0 ( x y ) = ( 0 x ) ( 0 y ) ) .
We can define a partial ordering ≤ on X as follows:
( x , y X ) x y x y = 0 .
A nonempty subset S of a B C K / B C I -algebra X is called a subalgebra of X if x y S for all x , y S . A subset I of a B C K / B C I -algebra X is called an ideal of X if it satisfies the following conditions:
0 I ,
( x , y X ) x y I , y I x I .
An ideal I of a B C I -algebra X is said to be closed if
( x X ) ( x I 0 x I ) .
We refer the reader to the books [6,7] for further information regarding B C K / B C I -algebras.

3. Quasi-Valuation Maps on BCK/BCI-Algebras

In what follows, let X denote a B C K / B C I -algebra unless otherwise specified.
Definition 1 ([5]).
By a quasi-valuation map of X based on an ideal (briefly I-quasi-valuation map of X), we mean a mapping f : X R which satisfies the conditions
f ( 0 ) = 0 ,
( x , y X ) f ( x ) f ( x y ) + f ( y ) .
The I-quasi-valuation map f is called an I-valuation map of X if
( x X ) ( f ( x ) = 0 x = 0 ) .
Lemma 1 ([5]).
For any I-quasi-valuation map f of X , we have the following assertions:
(1)
f is order reversing.
(2)
f ( x y ) + f ( y x ) 0 for all x , y X .
(3)
f ( x y ) f ( x z ) + f ( z y ) for all x , y , z X .
Corollary 1.
Every quasi-valuation map f of a B C K -algebra X satisfies:
( x X ) ( f ( x ) 0 ) .
Theorem 1.
For any ideal I of X, define a map
f I : X R , x 0 if x I , t otherwise ,
where t is a negative number in R . Then, f I is an I-quasi-valuation map of X. Moreover, f I is an I-valuation map of X if and only if I is the trivial ideal of X (i.e., I = { 0 } ).
Proof. 
Straightforward. ☐
Theorem 2.
If f is an I-quasi-valuation map of X, then the set
A f : = { x X f ( x ) 0 }
is an ideal of X.
Proof. 
Obviously 0 A f . Let x , y X be such that x y A f and y A f . Then, f ( x y ) 0 and f ( y ) 0 . It follows from (10) that f ( x ) f ( x y ) + f ( y ) 0 and so that x A f . Therefore A f is an ideal of X. ☐
Note that if an ideal of a B C I -algebra X is of finite order, then it is a closed ideal of X, and every ideal of a B C K -algebra X is a closed ideal of X (see [6]). Hence, we have the following corollary.
Corollary 2.
Let X be a finite B C I -algebra or a B C K -algebra. If f is an I-quasi-valuation map of X, then the set A f is a closed ideal of X.
Theorem 3.
If I is an ideal of X, then A f I = I .
Proof. 
We get A f I = { x X f I ( x ) 0 } = { x X x I } = I . ☐
Definition 2.
A real-valued function d on X × X is called a quasi-metric if it satisfies:
( x , y X ) d ( x , y ) 0 , d ( x , x ) = 0 ,
( x , y X ) d ( x , y ) = d ( y , x ) ,
( x , y , z X ) d ( x , z ) d ( x , y ) + d ( y , z ) .
The pair ( X , d ) is called the quasi-metric space.
Given a real-valued function f on X, define a mapping
d f : X × X R , ( x , y ) f ( x y ) + f ( y x ) .
Theorem 4.
If a real-valued function f on X is an I-quasi-valuation map of X, then d f is a quasi-metric on X × X .
The pair ( X , d f ) is called the quasi-metric space induced by f.
Proof. 
Using Lemma 1(2), we have d f ( x , y ) = f ( x y ) + f ( y x ) 0 for all ( x , y ) X × X . Obviously, d f ( x , x ) = 0 and d f ( x , y ) = d f ( y , x ) for all x , y X . Using Lemma 1(3), we get
d f ( x , y ) + d f ( y , z ) = ( f ( x y ) + f ( y x ) ) + ( f ( y z ) + f ( z y ) ) = ( f ( x y ) + f ( y z ) ) + ( f ( z y ) + f ( y x ) ) f ( x z ) + f ( z x ) = d f ( x , z )
for all x , y , z X . Therefore d f is a quasi-metric on X. ☐
Proposition 1.
Let f be an I-quasi-valuation map of a B C K -algebra X such that
( x X ) ( x 0 f ( x ) 0 ) .
Then, the quasi-metric space ( X , d f ) induced by f satisfies:
( x , y X ) ( d f ( x , y ) = 0 x = y ) .
Proof. 
Assume that d f ( x , y ) = 0 for x , y X . Then, f ( x y ) + f ( y x ) = 0 , and so f ( x y ) = 0 and f ( y x ) = 0 by Corollary 1. It follows from (15) that x y = 0 and y x = 0 . Hence x = y . ☐
We provide conditions for an I-quasi-valuation map to be an I-valuation map.
Theorem 5.
Let f be an I-quasi-valuation map of a B C I -algebra X such that A f is a closed ideal of X. If the quasi-metric d f induced by f satisfies the condition (16), then f is an I-valuation map of X.
Proof. 
Assume that f does not satisfy the condition (11). Then, there exists x X such that x 0 and f ( x ) = 0 . Thus, x A f , and so 0 x A f since A f is a closed ideal of X. Hence f ( 0 x ) 0 , which implies that
0 = f ( 0 ) f ( 0 x ) + f ( x ) = f ( 0 x ) 0 .
Thus, f ( 0 x ) = 0 , and so d f ( x , 0 ) = f ( x 0 ) + f ( 0 x ) = f ( x ) = 0 . It follows from (16) that x = 0 . Therefore, f is an I-valuation map of X. ☐
Since every ideal is closed in a B C K -algebra, we have the following corollary.
Corollary 3.
Given an I-quasi-valuation map f of a B C K -algebra X, if the quasi-metric d f induced by f satisfies the condition (16), then f is an I-valuation map of X.
Consider the B C I -algebra ( Z , , 0 ) and define a map f on Z as follows:
f k : Z R , x 0 if x = 0 , k x otherwise ,
where k is a negative integer. For any x Z \ { 0 } and y Z , we have f k ( x ) = k x and
f k ( x y ) + f k ( y ) = k x if either y = 0 or y = x , 2 k x otherwise .
It follows that f k ( x ) f k ( x y ) + f k ( y ) for all x , y Z , and so f k is an I-quasi-valuation map of ( Z , , 0 ) . It is clear that the set
A f k = { x Z f k ( x ) 0 } = { x Z x k } { 0 }
is an ideal of ( Z , , 0 ) which is not closed. Using Theorem 4, we know that d f k is a quasi-metric induced by f k and satisfies:
( x , y X ) ( d f k ( x , y ) = 0 x = y ) .
However, f k is not an I-valuation map of ( Z , , 0 ) since f k ( k ) = 0 and k 0 . This shows that if A f is not a closed ideal of X, then the conclusion of Theorem 5 is not true.
Proposition 2.
Given an I-quasi-valuation map f of X, the quasi-metric space ( X , d f ) satisfies:
(1)
d f ( x , y ) min { d f ( x a , y a ) , d f ( a x ) , d f ( a y ) } ,
(2)
d f ( x y , a b ) d f ( x y , a y ) + d f ( a y , a b ) ,
for all x , y , a , b X .
Proof. 
Let x , y , a , b X . Using (4), we have
( y a ) ( x a ) y x and ( x a ) ( y a ) x y .
Since f is order reversing, it follows that
f ( y x ) f ( ( y a ) ( x a ) ) and f ( x y ) f ( ( x a ) ( y a ) ) .
Thus,
d f ( x , y ) = f ( x y ) + f ( y x ) f ( ( y a ) ( x a ) ) + f ( ( x a ) ( y a ) ) = d f ( x a , y a ) .
Similarly, we get
d f ( x , y ) d f ( a x , a y ) .
Therefore, (1) is valid. Now, using Lemma 1(3) implies that
f ( ( x y ) ( a b ) ) f ( ( x y ) ( a y ) ) + f ( ( a y ) ( a b ) )
and
f ( ( a b ) ( x y ) ) f ( ( a b ) ( a y ) ) + f ( ( a y ) ( x y ) )
for all x , y , a , b X . Hence
d f ( x y , a b ) = f ( ( x y ) ( a b ) ) + f ( ( a b ) ( x y ) ) f ( ( x y ) ( a y ) ) + f ( ( a y ) ( a b ) ) + f ( ( a b ) ( a y ) ) + f ( ( a y ) ( x y ) ) f ( ( x y ) ( a y ) ) + f ( ( a y ) ( x y ) ) + f ( ( a b ) ( a y ) ) + f ( ( a y ) ( a b ) ) = d f ( x y , a y ) + d f ( a y , a b )
for all x , y , a , b X . Therefore, (2) is valid. ☐
Definition 3.
Let f be an I-quasi-valuation map of X. Define a relation θ f on X by
( x , y X ) ( x , y ) θ f f ( x y ) + f ( y x ) = 0 .
Theorem 6.
The relation θ f on X which is given in (17) is a congruence relation on X.
Proof. 
It is clear that θ f is an equivalence relation on X. Let x , y , u , v X be such that ( x , y ) θ f and ( u , v ) θ f . Then, f ( x y ) + f ( y x ) = 0 and f ( u v ) + f ( v u ) = 0 . It follows from Proposition 2 that
f ( ( x u ) ( y v ) ) + f ( ( y v ) ( x u ) ) = d f ( x u , y v ) d f ( x , y ) = f ( x y ) + f ( y x ) = 0 .
Hence, f ( ( x u ) ( y v ) ) + f ( ( y v ) ( x u ) ) = 0 , and so ( x u , y v ) θ f . Therefore, θ f is a congruence relation on X. ☐
Definition 4.
Let f be an I-quasi-valuation map of X and θ f be a congruence relation on X induced by f. Given x X , the set
x f : = { y X ( x , y ) θ f }
is called an equivalence class of x.
Denote by X f the set of all equivalence classes; that is,
X f : = { x f x X } .
Theorem 7.
Let f be an I-quasi-valuation map of X. Then, ( X f , , 0 f ) is a B C K / B C I -algebra where “⊙” is the binary operation on X f which is defined as follows:
( x f , y f X f ) x f y f = ( x y ) f .
Proof. 
Let X be a B C I -algebra. The operation ⊙ is well-defined since f is an I-quasi-valuation map of X. For any x f , y f , z f X f , we have
( ( x f y f ) ( x f z f ) ) ( z f y f ) = ( ( ( x y ) ( x z ) ) ( z y ) ) f = 0 f , ( x f ( x f y f ) ) y f = ( ( x ( x y ) ) y ) f = 0 f , x f x f = ( x x ) f = 0 f
.
Assume that x f y f = 0 f and y f x f = 0 f . Then, ( x y ) f = 0 f and ( y x ) f = 0 f , which imply that ( x y , 0 ) θ f and ( y x , 0 ) θ f . It follows from (1), (5), and (10) that
0 = f ( ( x y ) 0 ) + f ( 0 ( x y ) ) = f ( x y ) + f ( ( 0 x ) ( 0 y ) ) f ( x y ) + f ( 0 x ) f ( 0 y )
and
0 = f ( ( y x ) 0 ) + f ( 0 ( y x ) ) = f ( y x ) + f ( ( 0 y ) ( 0 x ) ) f ( y x ) + f ( 0 y ) f ( 0 x ) .
Hence, f ( x y ) + f ( 0 x ) f ( 0 y ) = 0 and f ( y x ) + f ( 0 y ) f ( 0 x ) = 0 , which imply that f ( x y ) + f ( y x ) = 0 . Hence, ( x , y ) θ f ; that is, x f = y f . Therefore, ( X f , , 0 f ) is a B C I -algebra. Moreover, if X is a B C K -algebra, then 0 x = 0 for all x X . Hence, 0 f x f = ( 0 x ) f = 0 f for all x f X f . Hence, ( X f , , 0 f ) is a B C K -algebra. ☐
The following example illustrates Theorem 7.
Example 1.
Let X = { 0 , a , b , c , d } be a set with the ∗-operation given by Table 1.
Then, ( X ; , 0 ) is a BCK-algebra (see [7]), and a real-valued function f on X defined by
f = 0 a b c d 0 4 9 0 11
is an I-quasi-valuation map of X (see [5]). It is routine to verify that
θ f = { ( 0 , 0 ) , ( a , a ) , ( b , b ) , ( c , c ) , ( d , d ) , ( 0 , c ) , ( c , 0 ) } ,
and X f = { 0 f , a f , b f , d f } is a B C K -algebra where 0 f = { 0 , c } , a f = { a } , b f = { b } , and d f = { d } .
Proposition 3.
Given an I-quasi-valuation map f of a B C I -algebra X, if A f is a closed ideal of X, then A f 0 f .
Proof. 
Let x A f . Then, 0 x A f since A f is a closed ideal, and so f ( x ) 0 and f ( 0 x ) 0 . It follows from (1) that
f ( 0 x ) + f ( x 0 ) = f ( 0 x ) + f ( x ) 0 ,
and so that f ( 0 x ) + f ( x 0 ) = 0 by using Lemma 1(2). Hence, ( 0 , x ) θ f ; that is, x 0 f . Therefore, A f 0 f . ☐
Corollary 4.
If f is an I-quasi-valuation map of a B C K -algebra X, then A f 0 f .
Proposition 4.
Let f be an I-quasi-valuation map of a B C I -algebra such that
( x X ) ( f ( x ) 0 ) .
Then, 0 f A f .
Proof. 
Let x 0 f . Then, ( 0 , x ) θ f , and s
f ( 0 x ) + f ( x ) = f ( 0 x ) + f ( x 0 ) = 0 .
It follows from (18) that f ( 0 x ) = 0 = f ( x ) . Hence, x A f , and therefore 0 f A f . ☐
Let I be an ideal of X and let η I be a relation on X defined as follows:
( x , y X ) ( ( x , y ) η I x y I , y x I ) .
Then, η I is a congruence relation on X, which is called the ideal congruence relation on X induced by I (see [6]). Denote by X / I the set of all equivalence classes; that is,
X / I : = { [ x ] I x X } ,
where [ x ] I = { y X ( x , y ) η I } . If we define a binary operation I on X / I by [ x ] I I [ y ] I = [ x y ] I for all [ x ] I , [ y ] I X / I , then ( X , I , [ 0 ] I ) is a B C K / B C I -algebra (see [6]).
Proposition 5.
If f is an I-quasi-valuation map of X, then η A f θ f .
Proof. 
Let x , y X be such that ( x , y ) η A f . Then, x y A f and y x A f , which imply that f ( x y ) 0 and f ( y x ) 0 . Hence, f ( x y ) + f ( y x ) 0 , and so f ( x y ) + f ( y x ) = 0 by using Lemma 1(2). Thus, ( x , y ) θ f . This completes the proof. ☐
Proposition 6.
If f is an I-quasi-valuation map of X such that A f = X , then θ f η A f .
Proof. 
Let x , y X be such that ( x , y ) θ f . Then, f ( x y ) + f ( y x ) = 0 , and so f ( x y ) = 0 and f ( y x ) = 0 by the condition A f = X . It follows that x y A f and y x A f . Hence, ( x , y ) η A f , and therefore θ f η A f . ☐
Theorem 8.
If I is an ideal of X, then η I = θ f I .
Proof. 
Let x , y X be such that ( x , y ) η I . Then, x y I and y x I . It follows that f I ( x y ) = 0 and f I ( y x ) = 0 . Hence, f I ( x y ) + f I ( y x ) = 0 , and thus ( x , y ) θ f I .
Conversely, let ( x , y ) θ f I for x , y X . Then, f I ( x y ) + f I ( y x ) = 0 , which implies that f I ( x y ) = 0 and f I ( y x ) = 0 since f I ( x ) 0 for all x X . Hence, x y I and y x I ; that is, ( x , y ) η I . This completes the proof. ☐
Corollary 5.
If f is an I-quasi-valuation map of X, then η A f = θ f A f .
Theorem 9.
For any two different I-quasi-valuation maps f and g of X, if 0 f = 0 g , then θ f and θ g coincide, and so X f = X g .
Proof. 
Let x , y X be such that ( x , y ) θ f . Then, ( x y , 0 ) = ( x y , y y ) θ f , and so x y 0 f . Similarly, we have y x 0 f . It follows from 0 f = 0 g that x g y g = ( x y ) g = 0 g and y g x g = ( y x ) g = 0 g . Hence, x g = y g , and so ( x , y ) θ g . Similarly, we can verify that if ( x , y ) θ g , then ( x , y ) θ f . Therefore, θ f and θ g coincide and so X f = X g . ☐
Theorem 10.
Let I be an ideal of X and let f be an I-quasi-valuation map of X such that 0 f I . If we denote
I f : = { x f x I } ,
then the following assertions are valid.
(1)
( x X ) ( x I x f I f ) .
(2)
I f is an ideal of X f .
Proof. 
(1) It is clear that if x I , then x f I f . Let x X be such that x f I f . Then, there exists y I such that x f = y f . Hence, ( x , y ) θ f , and so ( x y , 0 ) = ( x y , y y ) θ f . It follows that x y 0 f I and so that x I .
(2) Clearly, 0 f I f since 0 I . Let x , y X be such that x f y f I f and y f I f . Then, ( x y ) f = x f y f I f , and so x y I and y I by (1). Since I is an ideal of X, it follows that x I and so that x f I f . Therefore, I f is an ideal of X f . ☐
Theorem 11.
For any I-quasi-valuation map f of X, if J is an ideal of X f , then the set
J : = { x X x f J }
is an ideal of X containing 0 f .
Proof. 
It is obvious that 0 0 f J . Let x , y X be such that x y J and y J . Then, y f J and x f y f = ( x y ) f J . Since J is an ideal of X f , it follows that x f J (i.e., x J ). Therefore, J is an ideal of X. ☐
Let I ( X f ) denote the set of all ideals of X f , and let I ( X , f ) denote the set of all ideals of X containing 0 f . Then, there exists a bijection between I ( X f ) and I ( X , f ) ; that is, ψ : I ( X f ) I ( X , f ) , I I f is a bijection.
Proposition 7.
Let φ : X Y be a homomorphism of B C K / B C I -algebras. If f is an I-quasi-valuation map of Y, then the composition f φ of f and φ is an I-quasi-valuation map of X.
Proof. 
We have ( f φ ) ( 0 ) = f ( φ ( 0 ) ) = f ( 0 ) = 0 . For any x , y X , we get
( f φ ) ( x ) = f ( φ ( x ) ) f ( φ ( x ) φ ( y ) ) + f ( φ ( y ) ) = f ( φ ( x y ) ) + f ( φ ( y ) ) = ( f φ ) ( x y ) + ( f φ ) ( y ) .
Hence, f φ is an I-quasi-valuation map of X. ☐
Theorem 12.
Let φ : X Y be an onto homomorphism of B C K / B C I -algebras. If f is an I-quasi-valuation map of Y, then X f φ and Y f are isomorphic.
Proof. 
Define a map ζ : X f φ Y f by ζ ( x f φ ) = φ ( x ) f for all x X . If we let x f φ = a f φ for a , x X , then
0 = ( f φ ) ( x a ) + ( f φ ) ( a x ) = f ( φ ( x a ) ) + f ( φ ( a x ) ) = f ( φ ( x ) + φ ( a ) ) + f ( φ ( a ) φ ( x ) ) ,
which implies that ζ ( x f φ ) = φ ( x ) f = φ ( a ) f = ζ ( a f φ ) . Hence, ζ is well-defined. For any a , x X , we have
ζ ( x f φ a f φ ) = ζ ( ( x a ) f φ ) = φ ( x a ) f = ( φ ( x ) φ ( a ) ) f = φ ( x ) f φ ( a ) f = ζ ( x f φ ) ζ ( a f φ ) .
This shows that ζ is a homomorphism. For any y f in Y f , there exists x X such that φ ( x ) = y , since φ is surjective. It follows that ζ ( x f φ ) = φ ( x ) f = y f . Thus, ζ is surjective. Suppose that ζ ( x f φ ) = ζ ( a f φ ) for any x f φ , a f φ X f φ . Then, φ ( x ) f = φ ( a ) f , and so
( f φ ) ( x a ) + ( f φ ) ( a x ) = f ( φ ( x a ) ) + f ( φ ( a x ) ) = f ( φ ( x ) φ ( a ) ) + f ( φ ( a ) φ ( x ) ) = 0 .
Hence, x f φ = a f φ . This shows that ζ is injective, and therefore X f φ and Y f are isomorphic. ☐
Theorem 13.
Given an I-quasi-valuation map f of X, the following assertions are valid.
(1)
The map π : X X f , x x f is an onto homomorphism.
(2)
For each I-quasi-valuation map g of X f , there exist an I-quasi-valuation map g of X such that g = g π .
(3)
If A f = X , then the map
f : X f R , x f f ( x )
is an I-quasi-valuation map of X f .
Proof. 
(1) and (2) are straightforward.
(3) Assume that x f = y f for x , y X . Then, f ( x y ) + f ( y x ) = 0 , which implies from the assumption that f ( x y ) = 0 = f ( y x ) . Since x ( x y ) y for all x , y X , we get f ( y ) f ( x ( x y ) ) . It follows that
f ( x ) f ( x ( x y ) ) + f ( x y ) f ( x y ) + f ( y ) f ( y ) .
Similarly, we show that f ( x ) f ( y ) , and so f ( x ) = f ( y ) ; that is, f ( x f ) = f ( y f ) . Therefore, f is well-defined. Now, we have f ( 0 f ) = f ( 0 ) = 0 and
f ( x f ) = f ( x ) f ( x y ) + f ( y ) = f ( ( x y ) f ) + f ( y f ) = f ( x f y f ) + f ( y f ) .
Therefore, f is an I-quasi-valuation map of X f . ☐

4. Conclusions

Quasi-valuation maps on B C K / B C I -algebras were studied by Song et al. in [5]. The aim of this paper was to study the quotient structures of B C K / B C I -algebras induced by quasi-valuation maps. We have described relations between I-quasi-valuation maps and ideals in B C K / B C I -algebras. We have induced the quasi-metric space by using an I-quasi-valuation map of a B C K / B C I -algebra, and have investigated several properties. We have considered relations between the I-quasi-valuation map and the I-valuation map, and have provided conditions for an I-quasi-valuation map to be an I-valuation map. We have used I-quasi-valuation maps to introduce a congruence relation, and then constructed the quotient structures with related properties. We have established isomorphic quotient B C K / B C I -algebras. In the future, from a purely mathematical standpoint, we will apply the concepts and results in this article to related algebraic structures, such as B C C -algebras (see [8]), pseudo B C I -algebras (see [9,10]), and so on. From an application standpoint, we will try to find the possibility of extending our proposed approach to some decision-making problem, mathematical programming, medical diagnosis, etc.

Acknowledgments

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

Author Contributions

All authors contributed equally and significantly to the study and preparation of the manuscript. They have read and approved the final article.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Imai, Y.; Iséki, K. On axiom systems of propositional calculi, XIV. Proc. Jpn. Acad. 1966, 42, 19–22. [Google Scholar] [CrossRef]
  2. Iséki, K. An algebra related with a propositional calculus. Proc. Jpn. Acad. 1966, 42, 26–29. [Google Scholar] [CrossRef]
  3. Iséki, K. On BCI-algebras. Math. Semin. Notes 1980, 8, 125–130. [Google Scholar]
  4. Iséki, K.; Tanaka, S. An introduction to theory of BCK-algebras. Math. Japonica 1978, 23, 1–26. [Google Scholar]
  5. Song, S.Z.; Roh, E.H.; Jun, Y.B. Quasi-valuation mapd on BCK/BCI-algebras. Kyungpook Math. J. 2015, 55, 859–870. [Google Scholar] [CrossRef]
  6. Huang, Y.S. BCI-Algebra; Science Press: Beijing, China, 2006. [Google Scholar]
  7. Meng, J.; Jun, Y.B. BCK-Algebras; Kyungmoon Sa Co.: Seoul, Korea, 1994. [Google Scholar]
  8. Dudek, W.A.; Zhang, X.H. On atoms in BCC-Algebras. Discuss. Math. Algebra Stoch. Methods 1995, 15, 81–85. [Google Scholar]
  9. Zhang, X.H. Fuzzy anti-grouped filters and fuzzy normal filters in pseudo BCI-Algebras. J. Intell. Fuzzy Syst. 2017, 33, 1767–1774. [Google Scholar] [CrossRef]
  10. Zhang, X.H.; Park, C.; Wu, S.P. Soft set theoretical approach to pseudo BCI-algebras. J. Intell. Fuzzy Syst. 2018, 34, 559–568. [Google Scholar] [CrossRef]
Table 1. ∗-operation.
Table 1. ∗-operation.
0abcd
000000
aa00a0
bbb0b0
cccc0c
ddddd0

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top