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: hylinin@163.com

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

,,

yzX

0xy 0yx

, the following conditions:

1) BCL-1: ;

0xx

2) BCL-2: and

imply

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 ,,

yz

0xx

X

0yx

,

1) BCL-1: ;

2) BCL-2: and

0xy

imply

y

;

3)

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

yz X

,

1) BCL+-1: 1xx

1xy ;

2) BCL+-2: 1yx and imply

y

;

3) BCL+-3:

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

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