1. Introduction

Imai and Iski introduced two classes of abstract algebras: BCK-algebras and BCI-algebras ^[2] and ^[3]. The class of BCK-algebras is a proper subclass of the class of BCI-algebras. It is known that the notion of BCI-algebras is a generalization of BCK-algebras. J. Neggers and H. S. Kim ^[5] introduced the class of d-algebras which is a generalization of BCK-algebras, and investigated relations between d-algebras and BCK-algebras.

A partial multiplier on a commutative semigroup () has been introduced in ^[4] as a function F from a nonvoid subset DF of A into A such that for all The notion of multipliers on lattices was introduced and studied by ^{[6, 7]} and it was generalized to the partial multipliers on partially ordered sets in ^{[8, 9]}. Muhammad Anwar Chaudhry and Faisal Ali defined the notion of multipliers on d-algebras in ^[1].

In this paper the notion of symmetric bi-multipliers in d-algebras are given and properties of these multipliers are researched. Also, kernels and set of fixed points of a d-algebra are characterized by symmetric bi-multipliers.

2. Preliminaries

Definition 2.1. ^[5] A d-algebra is a non-empty set with a constant 0 and a binary operation denoted by * satisfying the following axioms for all

(I)

(II)

(III)

Definition 2.2. ^[5] Let S be a non-empty subset of a d-algebra X, then S is called subalgebra of X if for all

Definition 2.3. Let X be a d-algebra and I be a subset of X, then I is called an ideal of X if it satisfies the following conditions:

(1)

(2)

Definition 2.4. Let X be a d-algebra and I be a non-empty subset of X, then I is called a d-ideal of X if it satisfies the following conditions:

(1)

(2) and imply From condition (2) it is obvious that for

3. Symmetric Bi-multipliers on d-algebras

The following Definition introduces the notion of symmetric bi-multiplier for a d-algebra. In what follows, let X denote a d-algebra unless otherwise specified.

Definition 3.1. Let X be a d-algebra. A mapping is called symmetric if for all

Definition 3.2. Let X be a d-algebra and let be a symmetric mapping. We call f a symmetric bi-multiplier on X if it satisfies;

Example 3.1. Let and with the binary operation * defined by :

Then X is a d-algebra.

The mapping defined by

Then we can see that f is a symmetric bi-multiplier on X.

Remark 3.1. If X is a d-algebra with a binary operation *, then we can define a binary operation on X by;

Proposition 3.3. Let X be a d-algebra and f be the symmetric bi-multiplier on X. Then the followings hold for all :

i)

ii)

iii) If then

Proof: Let X be a d-algebra and f be the symmetric bi-multiplier on X.

i) By using the definition of symmetric bi-multiplier on X and (I) we have the following:

Therefore, .

ii) By i) we have

Therefore, we have and hence

iii) Let be elements in X and .

Therefore, we get and hence

Definition 3.4. ^[1] A d-algebra X is said to be positive implicative if

for all

Let S(X) be the collection of all symmetric bi-multipliers on X. It is clear that defined by for all and defined by for all are in S(X). Therefore, S(X) is not empty.

Definition 3.5. Let X be a positive implicative d-algebra and S(X) be the collection of all symmetric bi-multipliers on X. We define a binary operation * on S(X) by

and

Theorem 3.6. Let X be a positive implicative d-algebra. Then is a positive implicative d-algebra.

Proof: Let X be a positive implicative d-algebra and let Then

So,

Let Then

for al . So for all

Now let , we have

for all . So .

Let such that and This implies that and for all That is and which implies that for all Thus . Hence S(X) is a d-algebra.

Now we need to show that it is positive implicative. Let Then

for all Hence for all Therefore S(X) is an implicative d-algebra.

Definition 3.7. Let f be a symmetric bi-multiplier on X. We define Ker(f) by

for all .

Proposition 3.8. Let X be a d-algebra and f be the symmetric bi-multiplier on X. Then Ker(f) is a subalgebra of X.

Proof: Let X be a d-algebra and f be the symmetric bi-multiplier on X. Let Then we have and So Thus Therefore, is a subalgebra of X.

Definition 3.9. ^[1] A d-algebra X is called commutative if for all

Proposition 3.10. Let X be a commutative d-algebra satisfying and f be the symmetric bi-multiplier on X. If and then

Proof: Let and . Then we have and And then

Therefore, .

Definition 3.11. Let X be a d-algebra and f be the symmetric bi-multiplier on X. Then the set

for all is called the set of fixed points of f.

Proposition 3.12. Let X be a d-algebra and f be the symmetric bi-multiplier on X. Then is a subalgebra of X.

Proof: Let X be a d-algebra and f be the symmetric bi-multiplier on X.

Since is non-empty. Let . Then we have . Then

Therefore, . Hence, is a subalgebra of X.

Acknowledgements

The authors are highly grateful to the referees for their valuable comments and suggestions for the paper.

References

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