Y. H. LIU

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* 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 0xx

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

y

;

3)

*** ****

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.