International Journal of Mathematics and Mathematical Sciences

Volume 2013, Article ID 238490, 6 pages

http://dx.doi.org/10.1155/2013/238490

## On Symmetric Left Bi-Derivations in *BCI*-Algebras

^{1}Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia^{2}Department of Mathematics Education (and RINS), Gyeongsang National University, Jinju 660-701, Republic of Korea^{3}Department of Mathematics, Faculty of Science, Suleyman Demirel University, 32260 Isparta, Turkey

Received 15 February 2013; Accepted 30 May 2013

Academic Editor: Aloys Krieg

Copyright © 2013 G. Muhiuddin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The notion of *symmetric left bi-derivation* of a *BCI*-algebra *X* is introduced, and related properties are investigated. Some results on componentwise regular and *d*-regular *symmetric left bi-derivations* are obtained. Finally, characterizations of a *p*-semisimple *BCI*-algebra are explored, and it is proved that, in a *p*-semisimple *BCI*-algebra, *F* is a *symmetric left bi-derivation* if and only if it is a *symmetric bi-derivation*.

#### 1. Introduction

*BCK*-algebras and *BCI*-algebras are two classes of nonclassical logic algebras which were introduced by Imai and Iséki in 1966 [1, 2]. They are algebraic formulation of *BCK*-system and *BCI*-system in combinatory logic. Later on, the notion of *BCI*-algebras has been extensively investigated by many researchers (see [3–6], and references therein). The notion of a *BCI*-algebra generalizes the notion of a *BCK*-algebra in the sense that every *BCK*-algebra is a *BCI*-algebra but not vice versa (see [7]). Hence, most of the algebras related to the-norm-based logic such as MTL [8], BL, hoop, MV [9] (i.e lattice implication algebra), and Boolean algebras are extensions of *BCK*-algebras (i.e. they are subclasses of *BCK*-algebras) which have a lot of applications in computer science (see [10]). This shows that *BCK-*/*BCI*-algebras are considerably general structures.

Throughout our discussion,will denote a *BCI*-algebra unless otherwise mentioned. In the year 2004, Jun and Xin [11] applied the notion of derivation in ring and near-ring theory to *BCI*-algebras, and as a result they introduced a new concept, called a (regular) derivation, in *BCI*-algebras. Using this concept as defined they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of a-semisimple *BCI*-algebra. For a self-mapof a *BCI*-algebra, they defined a-invariant ideal and gave conditions for an ideal to be-invariant. According to Jun and Xin, a self mapis called a left-right derivation (briefly-derivation) ofifholds for all. Similarly, a self mapis called a right-left derivation (briefly-derivation) ofifholds for all. Moreover, ifis both- and-derivation, it is a derivation on. After the work of Jun and Xin [11], many research articles have appeared on the derivations of *BCI*-algebras and a greater interest has been devoted to the study of derivations in *BCI*-algebras on various aspects (see [12–17]).

Inspired by the notions of-derivation [18], left derivation [19], and symmetric bi-derivations [20, 21] in rings and near-rings theory, many authors have applied these notions in a similar way to the theory of *BCI*-algebras (see [12, 13, 17]). For instantce in 2005 [17], Zhan and Liu have given the notion of-derivation of *BCI*-algebras as follows: a self mapis said to be a left-right-derivation or--derivation ofif it satisfies the identityfor all. Similarly, a self mapis said to be a right-left-derivation or--derivation ofif it satisfies the identityfor all. Moreover, ifis both- and--derivation, it is said thatis an-derivation, whereis an endomorphism. In the year 2007, Abujabal and Al-Shehri [12] defined and studied the notion of left derivation of *BCI*-algebras as follows: a self mapis said to be a left derivation ofif satisfyingfor all. Furthermore, in 2011 [13], Ilbira et al. have introduced the notion of symmetric bi-derivations in *BCI*-algebras. Following [13], a mappingis said to be symmetric ifholds for all pairsA symmetric mappingis called left-right symmetric bi-derivation (briefly-symmetric bi-derivation) if it satisfies the identityfor all.is called right-left symmetric bi-derivation (briefly-symmetric bi-derivation) if it satisfies the identityfor all. Moreover, ifis both a- and a-symmetric bi-derivation, it is said thatis a symmetric bi-derivation on.

Motivated by the notion of *symmetric bi-derivations* [13] in the theory of *BCI*-algebras, in the present analysis, we introduced the notion of *symmetric left bi-derivations* on *BCI*-algebras and investigated related properties. Further, we obtain some results on componentwise regular and-regular *symmetric left bi-derivations*. Finally, we characterize the notion of-semisimple *BCI*-algebraby using the concept of *symmetric left bi-derivation* and show that, in a -semisimple *BCI*-algebrais a *symmetric left bi-derivation* if and only if it is a *symmetric bi-derivation*.

#### 2. Preliminaries

We begin with the following definitions and properties that will be needed in the sequel.

A nonempty setwith a constantand a binary operationis called a *BCI-algebra* if for allthe following conditions hold: (I)(II)(III)(IV)andimply

Define a binary relationonby lettingif and only if. Thenis a partially ordered set. A *BCI*-algebrasatisfyingfor all is called *BCK*-algebra.

A-algebrahas the following properties for all(a1). (a2)(a3)impliesand. (a4)(a5)(a6)(a7)implies

For a *BCI*-algebra, denote by(resp., ) the-part (resp., the-G part) of; that is,is the set of allsuch that(resp., ). Note that(see [22]). If, thenis called a*-semisimple **-algebra.* In a-semisimple-algebra, the following hold. (a8). (a9)for all. (a10). (a11)implies. (a12)implies. (a13)implies. (a14). (a15).

Letbe a-semisimple *BCI*-algebra. We define addition “+” asfor all. Thenis an abelian group with identityand. Conversely, letbe an abelian group with identity, and let. Thenis a-semisimple *BCI*-algebra andfor all(see [6]).

For a *BCI*-algebra we denote, in particular, andWe call the elements ofthe*-atoms* of. For any, let, which is called the *branch* ofwith respect to. It follows thatwheneverandfor alland all. Note thatwhich is the-semisimple part of, andis a-semisimple *BCI*-algebra if and only if(see [23, Proposition 3.2]). Note also that; that is,, which implies thatfor all. It is clear that, andandfor alland all. Letbe a symmetric mapping. Then for all, a mappingdefined byis called trace of[13]. For more details, refer to [3, 4, 6, 11, 22, 23].

#### 3. Symmetric Left Bi-Derivations

The following definition introduces the notion of *symmetric left bi-derivation* for a *BCI*-algebra.

*Definition 1. *A symmetric mappingis called a symmetric left bi-derivation ofif it satisfies the following identity:

*Example 2 (see [24]). * Consider a-semisimple *BCI*-algebrawith the following Cayley table:

Define a mappingby

It is routine to verify thatis a *symmetric left bi-derivation* of

Theorem 3. * Letbe a symmetric left bi-derivation ofThen *(1)(2)(3)(4)

*Proof. * (1) LetThen, and so
sinceHence.

(2) For anyimpliesand so

(3) By (2), we have. Then

(4) For anyand, we have
This completes the proof.

Using Theorem 3, we have the following corollary.

Corollary 4. *Letbe a symmetric left bi-derivation andbe the trace ofThen *(1)(2)

Theorem 5. * Letbe a symmetric left bi-derivation ofThen *(1)(2)(3)(4)

*Proof. * (1) Let. Then

(2) Supposefor all. It is clear that, for, we haveConversely let us assume thatthen by using Theorem 3(4), we have.

(3) For any, we have

(4) For any , we have
Thus, we can writefor anyThis completes the proof.

*Definition 6. *A *symmetric left bi-derivation *of a *BCI*-algebra is said to be *componentwise regular* if for all . In particular, is called -regular if .

Theorem 7. *Letbe a symmetric left bi-derivation of BCI-algebra. Thenis a BCK-algebra if and only ifis componentwise regular symmetric left bi-derivation. *

*Proof. * Supposeis a *BCK*-algebra. Then for any, we have
Henceis *componentwise regular*.

Conversely, letbe a *componentwise regular symmetric left bi-derivation*. Let for anybe such that. Then
But it is clear that
which is not possible asis a *componentwise regular symmetric left bi-derivation*. Thusis the unique-atom. Assume that for some, we have, then, so , which is a contradiction. Henceforth, for all, we haveimplying thereby,is a *BCK*-algebra.

This completes the proof.

Theorem 8. *Letbe a componentwise regular symmetric left bi-derivation of a BCI-algebraThen *(1)

*Bothandbelong to the same branch for all*(2)(3)

*Proof. * (1) For any, we get
sinceHenceand soObviously,.

(2) Sinceis componentwise regular,Then

(3) Sincefor allby (2), using (a3) we obtain
This completes the proof.

Next, we prove some results in a-semisimple *BCI*-algebra.

Theorem 9. *Letbe a symmetric left bi-derivation of a-semisimple BCI-algebra; one has the following assertions.*(1)(2)(3)

*Proof. * (1) Letbe a-semisimple *BCI*-algebra. Then for any, we have

(2) Let. Using (I), we have
These above inequalities can be rewritten as
Consequently, we get
Also, using Theorem 5(4) and (1), we obtain

Sinceis a-semisimple *BCI*-algebra, hence, by using (21) and (a12), the above (20) yields.

(3) We haveby Theorem 5(4). Further, on letting, we get that implies. Henceforth, which amounts to say that.

This completes the proof.

Theorem 10. *Letbe a-semisimple BCI-algebra. Thenis a symmetric left bi-derivation if and only if it is a symmetric bi-derivation on. *

*Proof. * Suppose thatis a *symmetric left bi-derivation* onFirst, we show thatis a *(r,l)-symmetric bi-derivation* on. Let. Using Theorem 9(1) and (a14), we have
Henceis a-*symmetric bi-derivation* on.

Again, we show thatis a *(l,r)-symmetric bi-derivation* on. Let. Using Theorem 9(1), (3) and (a15), we have

Conversely, suppose thatis a *symmetric bi-derivation* ofAsis a *(r,l)-symmetric bi-derivation* on, then for anyand using (a14), we have
Henceis a *symmetric left bi-derivation*. This completes the proof.

#### Acknowledgments

The authors are grateful to the anonymous referee(s) for a careful checking of the details and for helpful comments that improved the present paper. G. Muhiuddin and Abdullah M. Al-roqi were partially supported by the Deanship of Scientific Research, University of Tabuk, Ministry of Higher Education, Saudi Arabia.

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