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Advances in Pure Mathematics, 2012, 2, 59-61
http://dx.doi.org/10.4236/apm.2012.21012 Published Online January 2012 (http://www.SciRP.org/journal/apm)
On BCL+-Algebras
Yonghong Liu
School of Automation, Wuhan University of Technology, Wuhan, China
Email: [email protected]
Received October 11, 2011; revised November 24, 2011; accepted December 5, 2011
ABSTRACT
This paper presents the BCL+-algebras, which is derived the fundamental properties. Results are generalized with ver-
sion of BCL-algebras [5], using some unusual for a binary relation * and a constant 1 (one) in a non-empty set X, one
may take different axiom systems for BCL+-algebras.
Keywords: BCL-Algebra; BCL+-Algebra; Logic Algebra
1. Introduction
The BCK/BCI/BCH-algebra (see [1-4]) has been a major
issue, but BCL-algebra (see [5]) is a new algebra struc-
ture—and we started to grasp the properties. This paper
presents the BCL+-algebras, we show that under our
formulation, the BCL+-algebra is a variant of a BCL-
algebra. We can define by taking some axioms and im-
portant properties in this way for the BCL+-algebras.
A BCL-algebra may be defined as a non-empty set X
with a binary relation * and a constant 0 (zero) satisfying
the following axioms:
Definition 1.1. [5] An algebra
of type
is said to be a BCL-algebra if and only if for any
;,0X

2, 0
,,
x
yzX
0xy 0yx
, the following conditions:
1) BCL-1: ;
0xx
2) BCL-2: and
imply
x
y
;
3) BCL-3:







0xy zxzyzy x 
0
.
Such set X in Definition 1.1 is called the underlying set
of a BCL-algebra , which needs the following
theorem:
;,X
Theorem 1.1. [5] Algebra
;,0X of type
2, 0 is
a BCL-algebra if and only if it satisfies the following
conditions: for all ,,
x
yz
0xx
X
0yx
,
1) BCL-1: ;
2) BCL-2: and
0xy
imply
x
y
;
3)





x
yz xz y zyx
X

;,1X
,,
.
2. Main Result
The BCL+ product, denoted by *. We call the binary op-
eration * on X the * product on X, and the constant 1(one)
of X the unit element of X. For brevity we often write X
instead of . We begin with the following defini-
tion:
;,1
Definition 2.1. An algebra is called a BCL+-
algebra if it satisfies the following laws hold: for any
x
yz X
,
1) BCL+-1: 1xx
1xy ;
2) BCL+-2: 1yx and imply
x
y
;
3) BCL+-3:

x
yzxzy zyx
 

0,1, 2,3X
.
Such definition, clearly, the BCL+-algebra is a gener-
alization of the BCL-algebra, imply a BCL-algebra is a
BCL+-algebra, however, the converse is not true. We
illustrate with the next theorem.
Theorem 2.1. A BCL+-algebra is existent.
Proof. The proof of this Theorem 2.1 is not difficult
and uses only example. Let . Define an
operation * on X, which are given in Table 1.
;,1XThen
is a proper BCL+-algebra. It is easy to
verify that there are
BCI-1:


23 2113
11 3
13
30;
 



BCI-2:

223 3
21 3
13
30;




BCH-3: 1) The left side of the equation is
231111;

C
opyright © 2012 SciRes. APM
Y. H. LIU
60
Table 1. BCL+ operation.
* 0 1 2 3
0 1 0 0 0
1 0 1 3 3
2 3 1 1 1
3 1 3 3 1
2) The right side of the equation is

213 133


13 2 


 
21 3
13
 
2 32 3


;,1X

;,1X
y if 1,
,
xy
xy X


;X

;,1X
.
In the expression we see that 1 ≠ 3.
BCL-3:





 



23 121 3
11133 2
13 3
33
10.

 



BCL+-3: 1) The left side of the equation is



23 1
11
=1 3
=3;


2) The right side of the equation is

13 .
In the expression we see that BCL+-3 is valid. In fact,
it is not difficult to verity that BCL+-1 and BCL+-2 are
valid.
A BCL+-algebra is a partially ordered rela-
tion ≤ on X, now we obtain the following definition:
Definition 2.2. Suppose that is a BCL+-al-
gebra, the ordered relation if
if and onl
for all ,
xy (2.1)
then is partially ordered set and is an
algebra of partially ordered relation.
Corollary 2.1. Let every
x
X

;,1X
. Then 1(one) is
maximal element in a BCL+-algebra such that
1
x
imply . (2.2) 1x

0,,, ,1
Definition 2.3. A BCL+-algebra X is called proper
BCL+-algebra if X is not a BCL-algebra.
Example 2.1. Let
X
abc

;,1X
. We define an op-
eration * on X by Table 2.
In fact, it is not difficult to verify that
Table 2. BCL+ operation.
* 0 a b c 1
0 1 0 0 0 0
a a 1 1 c 1
b b a 1 c 1
c c b c 1 1
1 0 1 1 1 1
Theorem 2.2. Assume that is any a BCL+-
algebra. Then the following hold: for any

;,1X
,,
x
yz X
,
1xxy y1)

1
;
x
x1x imply 2)
;
1xy xzzy3)  
1xy ;
4) BCL+-2: 1yx and imply
x
y
.
Proof. Necessity. By BCL+-1 and 3), we obtain
 

111.xxy yxxyy
is a
BCL+-algebra.
 (2.3)
So, 1) holding.
By the same reasons, we derive
 

111 11111.xx xx
  (2.4)
Hence, 2) holding.
Sufficiency. It only needs to show BCL+-1. Substitut-
ing y for 1(one) in 1), we have
111xx

1
.
(2.5)
by y and x by z in 3), it follows
x
Replacing




111xxxxxx
 

. (2.6)
Using 2) and BCL+-1, we get


111 1xx xx
 . (2.7)
Clearly, an application of (2.5) to (2.7) can give
111xx
 . (2.8)
Comparing (2.4) with (2.7) and using BCL+-2, we get
11xx
. (2.9)
Also, by 2) and 1), the following holds:

11xxxxxx

1
. (2.10)
x
x
Combining (2.9) and (2.10) with 4) create

;,1X
,,
.
So Theorem 2.2 is valid.
Theorem 2.3. An algebra is a BCL+-algebra
if and only if it satisfies the following conditions: for all
x
yz X
,
1) BCL+-1: 1xx
1xy ;
2) BCL+-2: 1yx and imply
x
y
;
Copyright © 2012 SciRes. APM
Y. H. LIU
Copyright © 2012 SciRes. APM
61


1;zy x3)





xy zxzy 

1
Using Theorem 2.2 with 4), we obtain
x
yx . 4)
11
x
xxx
. (2.14)
 
Proof. The proof is routine. Necessity. To prove 1). By
BCL+-3.
;,1X
is a BCL+-algebra. Hence



1
1
xx xx
xx




1 1.xx
1
(2.11) 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
Then 1) holding.
Sufficiency. Substituting
x
for y and x for z in 3),
by BCL+-3 and 1), it follows








11
11 1.



1
x
xxxxx
xx x
 

xxx 
(2.12)
[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
(Kob e Uni versi ty), 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.
x
Also, substituting 1

for x in (2.11), by BCL+-3
and 1), we have
[5] Y. H. Liu, “A New Branch of the Pure Algebra: BCL-
Algebras,” Advances in Pure Mathematics, Vol. 1, No. 5,
2011, pp. 297-299. doi:10.4236/apm.2011.15054




1
1.
x



11
11
xx
x
xx
xx

 

xxx
xx



(2.13)