**Journal of Mathematical Sciences and Applications**

##
Symmetric Bi-multipliers on *d-algebras*

**Tamer Firat**^{1}, **Şule Ayar Özbal**^{1,}

^{1}Department of Mathematics, Faculty of Science and Letter, Yaşar University, Izmir, Turkey

Abstract | |

1. | Introduction |

2. | Preliminaries |

3. | Symmetric Bi-multipliers on d-algebras |

Acknowledgements | |

References |

### Abstract

In this study, we introduce the notion of symmetric bimultipliers in *d**-**algebras* and investigate some related properties. Among others kernels and sets of fixed points of a* d**-**algebra* are characterized by symmetric bi-multipliers.

**Keywords:** d- algebras, multipliers, fixed set, kernel

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Tamer Firat, Şule Ayar Özbal. Symmetric Bi-multipliers on
*d-algebras*.*Journal of Mathematical Sciences and Applications*. Vol. 3, No. 2, 2015, pp 22-24. https://pubs.sciepub.com/jmsa/3/2/1

- Firat, Tamer, and Şule Ayar Özbal. "Symmetric Bi-multipliers on
*d-algebras*."*Journal of Mathematical Sciences and Applications*3.2 (2015): 22-24.

- Firat, T. , & Özbal, Ş. A. (2015). Symmetric Bi-multipliers on
*d-algebras*.*Journal of Mathematical Sciences and Applications*,*3*(2), 22-24.

- Firat, Tamer, and Şule Ayar Özbal. "Symmetric Bi-multipliers on
*d-algebras*."*Journal of Mathematical Sciences and Applications*3, no. 2 (2015): 22-24.

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### 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)

**De****fi****nition 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

**De****fi****nition 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.

**De****fi****nition 3.1.** Let *X* be a *d**-**algebra*. A mapping is called symmetric if for all

**De****fi****nition 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** ** de**fi**ned 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 de**fi**ne 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 al**l **:*

*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

**De****fi****nition 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*.

**De****fi****nition 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*.

**De****fi****nition 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, .

**De****fi****nition 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

[1] | M. A. Chaudhry, and F. Ali, Multipliers in d-Algebras, World Applied Sciences Journal 18 (11):1649-1653, 2012. | ||

In article | |||

[2] | K. Iski, On BCI-algebras Math. Seminar Notes, 8 (1980), pp. 125130. | ||

In article | |||

[3] | K. Iski, S. Tanaka An introduction to theory of BCK-algebras, Math. Japonica, 23 (1978), pp. 126. | ||

In article | |||

[4] | R. LARSEN, An Introduction to the Theory of Multipliers, Berlin: Splinger-Verlag, 1971. | ||

In article | |||

[5] | J. Neggers, and Kim H.S. On d-Algebras , Math. Slovaca, Co., 49 (1999), 19-26. | ||

In article | |||

[6] | G. SZASZ Derivations of Lattices, Acta Sci. Math. (Szeged) 37 (1975), 149-154. | ||

In article | |||

[7] | G. SZASZ Translationen der Verbande, Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961), 53-57. | ||

In article | |||

[8] | A. SZAZ, Partial Multipliers on Partiall Ordered Sets, Novi Sad J. Math. 32(1) (2002), 25-45. | ||

In article | |||

[9] | A. SZAZ AND J. TURI, Characterizations of Injective Multipliers on Partially Ordered Sets, Studia Univ. "BABE-BOLYAI" Mathematica XLVII(1) (2002), 105-118. | ||

In article | |||