Abstract and Applied Analysis

Volume 2012, Article ID 872784, 12 pages

http://dx.doi.org/10.1155/2012/872784

## On -Derivations of BCI-Algebras

Department of Mathematics, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Received 29 January 2012; Revised 19 April 2012; Accepted 20 April 2012

Academic Editor: Gerd Teschke

Copyright © 2012 G. Muhiuddin and Abdullah M. Al-roqi. 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

We introduce the notion of *t*-derivation of a BCI-algebra and investigate related properties. Moreover, we study *t*-derivations in a *p*-semisimple BCI-algebra and establish some results on *t*-derivations in a *p*-semisimple BCI-algebra.

#### 1. Introduction

The notion of BCK-algebra was proposed by Imai and Iséki in 1966 [1]. In the same year, Iséki introduced the notion of a BCI-algebra [2], which is a generalization of a BCK-algebra. A series of interesting notions concerning BCI-algebras were introduced and studied, several papers have been written on various aspects of these algebras [3–5]. Recently, in the year 2004 [6], Jun and Xin have applied the notion of derivation in BCI-algebras which is defined in a way similar to the notion of derivation in rings and near-rings theory which was introduced by Posner in 1957 [7]. In fact, the notion of derivation in ring theory is quite old and plays a significant role in analysis, algebraic geometry and algebra.

After the work of Jun and Xin (2004) [6], many research articles have appeared on the derivations of BCI-algebras in different aspects as follows: in 2005 [8], Zhan and Liu have given the notion of -derivation of BCI-algebras and studied -semisimple BCI-algebras by using the idea of regular -derivation in BCI-algebras. In 2006 [9], Abujabal and Al-Shehri have extended the results of BCI-algebras. Further, in the next year 2007 [10], they defined and studied the notion of left derivation of BCI-algebras and investigated some properties of left derivation in -semisimple BCI-algebras. In 2009 [11], Öztürk and Çeven have defined the notion of derivation and generalized derivation determined by a derivation for a complicated subtraction algebra and discussed some related properties. Also, in 2009 [12], Öztürk et al. have introduced the notion of generalized derivation in BCI-algebras and established some results. Further, they have given the idea of torsion free BCI-algebra and explored some properties. In 2010 [13], Al-Shehri has applied the notion of left-right (resp., right-left) derivation in BCI-algebra to B-algebra and obtained some of its properties. In 2011 [14], Ilbira et al. have studied the notion of left-right (resp., right-left) symmetric biderivation in BCI-algebras.

Motivated by a lot of work done on derivations of BCI-algebras and on derivations of other related abstract algebraic structures, in this paper we introduce the notion of -derivations on BCI-algebras and obtain some of its related properties. Further, we characterize the notion of -semisimple BCI-algebra by using the notion of *t*-derivation and show that if and are -derivations on , then is also a -derivation and . Finally, we prove that , where and are -derivations on a -semisimple BCI-algebra.

#### 2. Preliminaries

We review some definitions and properties that will be useful in our results.

*Definition 2.1 (see [2]). * Let be a set with a binary operation “” and a constant . Then is called a BCI algebra if the following axioms are satisfied for all : (i), (ii), (iii), (iv) and .

Define a binary relation on by letting if and only if . Then is a partially ordered set. A BCI-algebra satisfying for all , is called BCK-algebra (see [1]).

In any BCI-algebra for all , the following properties hold. (1). (2). (3). (4) implies . (5) and . A BCI-algebra is said to be associative if for all , the following holds: (6) [4]. Let be a BCI-algebra, we denote , the BCK-part of and by , the BCI-G part of . If , then is called a -semisimple BCI-algebra. In a -semisimple BCI-algebra , the following properties hold.(7).(8). (9) implies . (10). (11) implies that is left cancelable. (12) implies that is right cancelable.

*Definition 2.2 (see [6]). *A subset of a BCI-algebra is called subalgebra of if whenever .

For a BCI-algebra , we denote for all [6]. For more details we refer to [3, 5, 6].

#### 3. -Derivations in a BCI-Algebra/-Semisimple BCI-Algebra

The following definitions introduce the notion of -derivation for a BCI-algebra.

*Definition 3.1. *Let be a-BCI-algebra. Then for any , we define a self map by for all .

*Definition 3.2. *Let be a BCI-algebra. Then for any , a self map is called a left-right -derivation or --derivation of if it satisfies the identity for all .

Similarly, we get the following.

*Definition 3.3. *Let be a BCI-algebra. Then for any , a self map is called a right-left -derivation or --derivation of if it satisfies the identity for all .

Moreover, if is both a - and a --derivation on , we say that is a -derivation on .

*Example 3.4. *Let be a BCI-algebra with the following Cayley table:
For any , define a self map by for all . Then it is easily checked that is a -derivation of .

Proposition 3.5. *Let be a self map of an associative BCI-algebra . Then is a --derivation of .*

*Proof. *Let be an associative BCI-algebra, then we have

Proposition 3.6. *Let be a self map of an associative BCI-algebra . Then, is a --derivation of .*

*Proof. *Let be an associative BCI-algebra, then we have
Combining Propositions 3.5 and 3.6, we get the following Theorem.

Theorem 3.7. *Let be a self map of an associative BCI-algebra . Then, is a -derivation of .*

*Definition 3.8. *A self map of a BCI-algebra is said to be -regular if .

*Example 3.9. *Let be a BCI-algebra with the following Cayley table:

(i) For any , define a self map by
Then it is easily checked that is and --derivations of , which is not -regular.

(ii) For any , define a self map by
Then it is easily checked that is and --derivations of , which is -regular.

Proposition 3.10. *Let be a self map of a BCI-algebra . Then *(i)If is a --derivation of , then for all . (ii) If is a --derivation of , then for all if and only if is t-regular.

*Proof of (i). *Let be a --derivation of , then
But is trivial so (i) holds.

*Proof of (ii). * Let be a --derivation of . If then
thereby implying is -regular. Conversely, suppose that is -regular, that is , then we have
This completes the proof.

Theorem 3.11. *Let be a --derivation of a -semisimple BCI-algebra . Then the following hold: *(i)* for all . *(ii)* is one-one. *(iii)*If is t-regular, then it is an identity map. *(iv)*if there is an element such that , then is identity map. *(v)*if , then for all . *

*Proof of (i). *Let be a --derivation of a -semisimple BCI-algebra . Then for all , we have and so

*Proof of (ii). *Let , then by property (12), we have and so is one-one.

*Proof of (iii). *Let be -regular and . Then, so by the above part (i), we have and hence by property (9), we obtain for all . Therefore, is the identity map.

*Proof of (iv). *It is trivial and follows from the above part (iii).

*Proof of (v). *Let implying . Now,
Therefore, . This completes the proof.

*Definition 3.12. *Let be a -derivation of a BCI-algebra . Then, is said to be an isotone -derivation if for all .

*Example 3.13. *In Example 3.9(ii), is an isotone -derivation, while in Example 3.9(i), is not an isotone -derivation.

Proposition 3.14. *Let be a BCI-algebra and be a -derivation on . Then for all , the following hold: *(i)If , then is an isotone -derivation. (ii)If , then is an isotone -derivation.

*Proof of (i). *Let . If for all . Therefore, we have
Henceforth which implies that is an isotone -derivation.

*Proof of (ii). *Let . If for all . Therefore, we have
Thus, . This completes the proof.

Theorem 3.15. *Let be a -regular --derivation of a BCI-algebra . Then, the following hold: *(i)* for all . *(ii)* for all . *(iii)* for all . *(iv)* is a subalgebra of . *

*Proof of (i). *For any , we have .

*Proof of (ii). *Since for all , then and hence the proof follows.

*Proof of (iii). *For any , we have

*Proof of (iv). *Let . From (iii), we have implying and so . Therefore, . Consequently is a subalgebra of . This completes the proof.

*Definition 3.16. *Let be a BCI-algebra and let , be two self maps of . Then we define by for all .

*Example 3.17. *Let be a BCI algebra which is given in Example 3.4. Let and be two self maps on as defined in Example 3.9(i) and Example 3.9(ii), respectively.

Now, define a self map by
Then, it is easily checked that for all .

Proposition 3.18. *Let be a -semisimple BCI-algebra and let , be --derivations of . Then, is also a --derivation of .*

*Proof. *Let be a -semisimple BCI-algebra. and are --derivations of . Then for all , we get
Therefore, is a --derivation of .

Similarly, we can prove the following.

Proposition 3.19. *Let be a -semisimple BCI-algebra and let , be --derivations of . Then is also a --derivation of .*

Combining Propositions 3.18 and 3.19, we get the following.

Theorem 3.20. *Let be a -semisimple BCI-algebra and let , be -derivations of . Then, is also a -derivation of .*

Now, we prove the following theorem.

Theorem 3.21. *Let be a -semisimple BCI-algebra and let , be -derivations of . Then .*

*Proof. *Let be a -semisimple BCI-algebra. and , *-*derivations of . Suppose is a --derivation, then for all , we have
As is a --derivation, then
Again, if is a --derivation, then we have
But is a --derivation, then
Therefore from (3.18) and (3.20), we obtain
By putting , we get
Hence, . This completes the proof.

*Definition 3.22. *Let be a BCI-algebra and let , be two self maps of . Then we define by for all .

*Example 3.23. *Let be a BCI algebra which is given in Example 3.4. Let and be two self maps on as defined in Example 3.9(i) and Example 3.9(ii), respectively.

Now, define a self map by
Then, it is easily checked that for all .

Theorem 3.24. *Let be a -semisimple BCI-algebra and let , be -derivations of . Then .*

*Proof. *Let be a -semisimple BCI-algebra. and , -derivations of .

Since is a --derivation of , then for all , we have
But is a --derivation, so
Again, if is a --derivation of , then for all , we have
As is a --derivation, then
Henceforth from (3.25) and (3.27), we conclude
By putting , we get
Hence, . This completes the proof.

#### 4. Conclusion

Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. The theory of derivations of algebraic structures is a direct descendant of the development of classical Galois theory (namely, Suzuki [15] and Van der Put and Singer [16, 17]) and the theory of invariants. An extensive and deep theory has been developed for derivations in algebraic structures viz. BCI-algebras, -algebras, commutative Banach algebras and Galois theory of linear differential equations (see, e.g., Jun and Xin [6], Ara and Mathieu [18], Bonsall and Duncan [19], Murphy [20] and Villena [21] where further references can be found). It plays a significant role in functional analysis; algebraic geometry; algebra and linear differential equations.

In the present paper, we have considered the notion of -derivations in BCI-algebras and investigated the useful properties of the -derivations in BCI-algebras. Finally, we investigated the notion of -derivations in a -semisimple BCI-algebra and established some results on -derivations in a -semisimple BCI-algebra. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras [11], B-algebras [13], MV-algebras [22], d-algebras, Q-algebras and so forth. In future we can study the notion of -derivations on various algebraic structures which may have a lot of applications in different branches of theoretical physics, engineering and computer science. It is our hope that this work would serve as a foundation for the further study in the theory of derivations of BCK/BCI-algebras.

In our future study of -derivations in BCI-algebras, may be the following topics should be considered: (1)to find the generalized -derivations of BCI-algebras, (2)to find more results in -derivations of BCI-algebras and its applications, (3)to find the -derivations of B-algebras, Q-algebras, subtraction algebras, d-algebra and so forth.

#### Acknowledgments

The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and several useful suggestions. This research is supported by the Deanship of Scientific Research, University of Tabuk, Tabuk, Saudi Arabia.

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