Advances in Pure Mathematics, 2011, 1, 297-299
doi:10.4236/apm.2011.15054 Published Online September 2011 (http://www.SciRP.org/journal/apm)
A New Branch of the Pure Algebra: BCL-Algebras
Yonghong Liu
School of Automation, Wuhan University of Technology, Wuhan, China
Received May 18, 2011; revised June 23, 2011; accepted July 5, 2011
Abstract
The BCK/BCI/BCH-algebras finds general algebra system than Boolean algebras system. This paper pre-
sents a novel class of algebras of type (2, 0) called BCL-algebras. We found the BCL-algebras to be more
extensive class than BCK/BCI/BCH-algebras in the abstract algebra. The BCL-algebras as a class of logical
algebras are the algebraic formulations of the set difference together with its properties in set theory and the
propositional calculus in logical systems. It is important that the BCL-algebras play an independent role in
the axiom algebra system.
Keywords: Logic Algebra, BCK-Algebra, BCI-Algebra, BCH-Algebra, BCL-Algebra
1. Introduction
In [1,2], the BCK-algebras and BCI-algebras are abbre-
viated to two B-algebras. The former was raised in 1966
Y. Imai and K. Iséki, and the latter was primitives in the
same year due to K. Iséki [3]. In 1983, Q. P. Hu and X.
Li [4,5] defined a class of algebras of type (2, 0) called
BCH-algebras base on BCK-algebras and BCI-algebras.
In this paper we present the BCL-algebras, namely,
L-algebras of type (2, 0).
To begin with, let’s examine BCI-algebras and BCK-
algebras that we have all observed. In fact, BCK-alge-
bras are a special class of BCI-algebras; we have the fol-
lowing nice results.
Definition 1.1. [2] An algebra of type (2, 0)
is called a BCI-algebra if it satisfies the following condi-
tions: for any

;*,0X

,,
x
yz X

,
1) BCI-1:
 


*****0xy xzzy
 
** *0xxy y
;
2) BCI-2: ;
3) BCI-3: ;
*0xx
4) BCI-4: and imply
*xy0*0yx
x
y

;
5) BCI-5: imply .
*0 0x0x
Theorem 1.1. [3] Given a BCI-algebra X, then the fol-
lowing identity holds: for any x, y, ,
zX



**** 0xy zxzy (1)
Definition 1.2. [6] Given a BCI-algebra X, if it satis-
fies the condition
BCK-4: for all
0* 0x
x
X

(i.e., every element
x
X is positive), we call it the BCK-algebra.
Definition 1.3. [4] An algebra of type (2, 0)
is called a BCH-algebra if it satisfies the following con-
ditions: for any x, y,

;*,0X

zX

,
1) BCH-1: *0xx

;
2) BCH-2: *xy 0

and imply *0yx
x
y

;
3) BCH-3:




** **
x
yz
*0
xzy
.
2. BCL-Algebras
There are several axiom systems for BCL-algebras as
shown in the following.
Definition 2.1. An algebra of type (2, 0) is
said to be a BCL-algebra if and only if for any x, y, z in X,
the following conditions:

;*,0X

1) BCL-1: xx

;
2) BCL-2: *xy 0

and imply *0yx
x
y

;
3) BCL-3:


 




 
***z*** **0xyxzyzy x

.
Such set X in Definition 2.1 is called the underlying
set of a BCL-algebra


0;*,X.
Definition 2.2. Let


;*,0X is a BCL-algebra. A
binary relation ≤ on X by which x ≤ y if and only if
*xy 0

for any x, y, zX

, we call the BCL-ordering
≤ is a partial ordering on X.
Definition 2.3. Let
x
y

if and only if *0xy

,
the Definition 2.1 can then be written as
1) BCL-1*:
x
x

;
2) BCL-2*:
x
y

and
y
x imply
x
y;
3) BCL-3*:




**



* ****

x
yzxzy zyx.
Definition 2.4. From the class of all BCL-algebra, we
denote by BCL.
Theorem 2.1. 1) Any a BCK-algebra is a BCL-alge-
bra;

Y. H. LIU 298

2) Any a BCI-algebra is a BCL-algebra;
3) Any a BCH-algebra is a BCL-algebra.
Proof: For Theorem 2.1 1), 2) and 3), notes how the
basic fact- BCK-algebraic class BCI-algebraic class
BCH-algebraic class, we only need to prove the fol-
lowing result.


Let be a BCH-algebra, suppose that x, y,
,

;*,0X
0*zX
y
y and*
x
yz, using BCH-1 and BCH-
3 then
   

 
    
   

 

 

****** **
***** **
0* *****
0*0 ****
0***
0* *
0*0 0
x
yz xzyzyx
zzxy zzxy
zzxy xy
xx y y
xy y
yy






(2)
Therefore, be a BCL-algebra.

;*,0XX


Theorem 2.2. Let be a BCL-algebra, if

;*,0
 
** **
x
yz xzy for any x, y, (3) zX
Then the BCL-algebra is a BCH-algebra.
Proof: Let


;*,0X
0*
be a BCH-algebra, suppose that
x, y, , zX
y
y and *
x
yz, using BCH-1 and
BCH-3, by Theorem 2.1(3), we have

** 0xy z

.
Similarly, suppose that x, y,
zX 0*zz

and
*
x
zy, using BCH-1 and BCH-3. We then have
   

 
   
   

 

 

***** ***
***** **
0**** **
0*0 ****
0***
0* *
0*0 0
x
zy xyzyzx
yyxzyyx z
yyxz x z
xx z z
xz z
zz






(4)
Hence . Combining this with equation is
, we obtain equation (3). Finally, the Theo-
rem 2.2 is proved.

** 0xz y
0
z

**xy
As this Theorem 2.1 and 2.2 indicate, we summarize
these results in the following contains.
Corollary 2.1. BCK-algebraic class BCI-alge-
braic class  BCH-algebraic class BCL-algebraic
class A (2, 0), where A (2, 0) is algebraic class of
type (2, 0).



Theorem 2.3. Let


;*,0X be a BCL-algebra. Then
the following hold: for any x, y, ,
zX
1) BCL-3*:
 

  
*** ****
x
yzxzy zyx;
2) *0xy

if and only if
x
y.
Proof: Assume that


;*,0X is a BCL-algebra, then
the BCL-ordering ≤ is a partial ordering on X. By defini-
tion of ≤, (2) is valid. Also, BCL-3 and (2) imply (1).
Conversely, assume that ≤ is a partial ordering on X, and
satisfying (1) and (2). Also, by the reflexivity of ≤, we
see that
x
x

, then (2) implies . *0xx
Moreover, if *xy 0

and , then *0yx
x
y

and
y
x

by (2), and so the anti-symmetry of ≤ gives
x
y

.
Therefore


;*,0X is a BCL-algebra.
Definition 2.5. If a BCL-algebra

is not
BCK/BCI/BCH-algebra, it is called proper BCL-algebra.



;*,0X
Theorem 2.4. There is an algebra of type
(2, 0) which proper BCL-algebra is exists.

;*,0X
Proof: The proof of this Theorem 2.4 is not difficult
and uses only example. Let . Define an
operation * on X, we find

0,1, 2,3X
* 0 1 2 3
0 0 0 0 0
1 1 0 3 1
2 2 3 0 2
3 3 0 0 0
Then


;*,0X is a proper BCL-algebra. It is easy to
verify that there are
BCI-1:










2*3 *2*1*1*32*3 *12*130
.
BCH-3: 1) On the left side of equation is


2*3 *1
2*1 3

;
2) On the right side of equation is

2*1 *33*3
0

.
In the expression we see that . In fact, it is not
difficult to verity that BCL-1, BCL-2 and BCL-3 are
valid.
30
Example 2.1. The set

0,1, ,

X
ab. Define a binary
operation * on X given by the following * multiplication
table:
* 0 1 a b
0 0 0 0 0
1 1 0 b 1
a a b 0 a
b b a 0 0
forms a proper BCL-algebra.
Example 2.2. Let


0,1, 2, 3X. We define a binary
operation * on X by

Y. H. LIU 299
* 0 1 2 3
0 0 1 2 3
1 1 0 1 2
2 2 0 0 0
3 3 2 0 0
It is not difficult to verify that is a proper
BCL-algebra.

;*,0X



Fact 2.1. The example above implies BCL-algebra
theory is independent algebra system.
Theorem 2.5. 1) A proper class is composed of all
BCL-algebras; it is called a BCL-algebraic class;
2) All BCK/ BCI /BCH-algebraic class are proper
subclass of BCL-algebraic class.
Proof: 1) By BCI /BCH-algebraic class is a proper
class, and by Theorem 2.3 tells us that BCL-algebraic
class is a proper class.
2) By Theorem 2.4 it is obvious that BCK/BCI
/BCH-algebraic class is a proper subclass of BCL-alge-
braic class.
Remark 2.1. It is not necessary that the algebra
of all type (2, 0) are BCL-algebras. Let’s work
out an example.

;*,0X
Example 2.3. Let . Define a binary op-
eration * on X by

0,1, 2X
* 0 1 2
0 0 1 2
1 0 0 1
2 2 1 0
Then (X; *, 0) is not BCL-algebra, we obtain



 


1*2*1 *1*1 *2*1*2*12*020. (5)
BCL-3 does not hold for the algebra of type (2, 0). But
1*1 = 0, it is satisfies BCL-1 and BCL-2.
Theorem 2.6. Let


;*,0X be a BCL-algebra, we
have the following relations: for any x, y, ,
zX
1) BCI-2: ;


** *0xxy y
*0 0xx
2) BCI-5: imply .
0
Proof: The theorem follows immediately from Defini-
tion 1.1 above.
Theorem 2.7. A BCL-algebra is called
proper BCL-algebra if and only if it not satisfies BCI-1
and BCH-3.

;*,0X

Proof: Necessity. If


;*,0X satisfies BCI-1 and
BCH-3. By Theorem 2.6, it satisfies BCI-2 and BCI-5,
and then we obtain a BCI-algebra and a BCH-algebra,
contradicting conditions of a proper BCL-algebra.
Sufficiency. Since


;*,0X not satisfies BCI-1 and
BCH-3. Therefore, it is not BCI-algebra and
BCH-algebra but proper BCL-algebra.
Theorem 2.8. Algebra


;*,0X of type (2, 0) is a
BCL-algebra if and only if it satisfies the following con-
ditions: for all x, y, zX

,
1) BCL-1: *0xx

;
2) BCL-2: *xy 0

and imply
*0yx
x
y

;
3)









*** ****
x
yzxzy zyx. 
Proof: We shall only prove the Theorem 2.8 3), by
Theorem 2.1, Theorem 2.3 and Definition 1.2. So 3) is
valid.
3. Conclusions
Taking theory of sets and propositional calculus as the
backdrop, the new study suggests that the BCL-algebras
are an important algebra in the axiom algebra system,
which delves into generalizations of difference opera-
tions and characteristic.
4. References
[1] Y. Imai and K. Iséki, “On Axiom System of Propositional
Calculi XIV,” Proceedings of the Japan Academy, Vol.
42, No. 1, 1966, pp. 19-22. doi:10.3792/pja/1195522169
[2] K. Iséki, “An Algebra Related with a Propositional Cal-
culus,” Proceedings of the Japan Academy, Vol. 42, No.
1, 1966, pp. 26-29. doi:10.3792/pja/1195522171
[3] K. Iseki, “On BCI-Algebras,” Mathematics Seminar Notes
(Kobe University), Vol. 8, No. 1, 1980, pp. 125-130.
[4] Q. P. Hu and X. Li, “On BCH-Algebras,” Mathematics
Seminar Notes (Kobe University), Vol. 11, No. 2, 1983,
pp. 313-320.
[5] Q. P. Hu and X. Li, “On Proper BCH-Algebras,” Mathe-
matica Japonica, Vol. 30, No.4, 1985, pp. 659-661.
[6] Y. S. Huang, “BCI-Algebra,” Science Press, Beijing,
2006, p. 21.