A New Branch of the Pure Algebra: Bcl-algebras

The BCK/BCI/BCH-algebras finds general algebra system than Boolean algebras system. This paper presents 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.


Introduction
In [1,2], the BCK-algebras and BCI-algebras are abbreviated 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 BCKalgebras that we have all observed.In fact, BCK-algebras are a special class of BCI-algebras; we have the following nice results.
Definition 1.1.[2] An algebra of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: for any [3] Given a BCI-algebra X, then the following identity holds: for any x, y, Definition 1.2.[6] Given a BCI-algebra X, if it satisfies the condition BCK-4: for all 0* 0 x  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 conditions: for any x, y, x z y .

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: 3 From the class of all BCL-algebra, we denote by BCL.
Theorem 2.1.1) Any a BCK-algebra is a BCL-algebra;  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 following result.

Let
be a BCH-algebra, suppose that x, y, ,  ;*, 0 X 0* z X  y y  and * x y z  , using BCH-1 and BCH-3 then Therefore, be a BCL-algebra.
 ;*, 0 Similarly, suppose that x, y, z X 0* z  z  and * x z y  , using BCH-1 and BCH-3.We then have Hence .Combining this with equation is , we obtain equation (3).Finally, the Theorem 2.2 is proved.
As this Theorem 2.1 and 2.2 indicate, we summarize these results in the following contains.2) * 0 x y  if and only if x y  .
Proof: Assume that   ;*, 0 X is a BCL-algebra, then the BCL-ordering ≤ is a partial ordering on X.By definition 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  ;*, 0 X Proof: The proof of this Theorem 2.4 is not difficult and uses only example.Let . Define an operation * on X, we find 0,1,2,3 is a proper BCL-algebra.It is easy to verify that there are BCH-3: 1) On the left side of equation is   2) On the right side of equation is   In the expression we see that .In fact, it is not difficult to verity that BCL-1, BCL-2 and BCL-3 are valid.
. Define a binary operation * on X given by the following * multiplication table: . We define a binary operation * on X by It is not difficult to verify that is a proper BCL-algebra.

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 operations and characteristic.
z X  Then the BCL-algebra is a BCH-algebra.Proof: Let   ;*, 0 X 0* be a BCH-algebra, suppose that x, y, , z X  y y  and * x y z  , using BCH-1 and BCH-3, by Theorem 2.1(3), we have   * * 0 x y z  .