Journal of Applied Mathematics

Volumeย 2012, Article IDย 626781, 8 pages

http://dx.doi.org/10.1155/2012/626781

## Coderivations of Ranked Bigroupoids

^{1}Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{2}Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea^{3}School of Digital Mechanics, Ulsan College, Nam-Gu Ulsan 680-749, Republic of Korea

Received 29 May 2012; Revised 7 August 2012; Accepted 7 August 2012

Academic Editor: Hak-Keungย Lam

Copyright ยฉ 2012 Young Bae Jun 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 (co)derivations of ranked bigroupoids is discussed by Alshehri et al. (in press), and their generalized version is studied by Jun et al. (under review press). In particular, Jun et al. (under review press) studied coderivations of ranked bigroupoids. In this paper, the generalization of coderivations of ranked bigroupoids is discussed. The notion of generalized coderivations in ranked bigroupoids is introduced, and new generalized coderivations of ranked bigroupoids are obtained by combining a generalized self-coderivation with a rankomorphism. From the notion of ()-derivation, the existence of a rankomorphism of ranked bigroupoids is established.

#### 1. Introduction

Several authors [1โ4] have studied derivations in rings and near-rings. Jun and Xin [5] applied the notion of derivation in ring and near-ring theory to -algebras, and as a result they introduced a new concept, called a (regular) derivation, in -algebras. Alshehri [6] applied the notion of derivations to incline algebras. Alshehri et al. [7] introduced the notion of ranked bigroupoids and discussed -self-(co)derivations. Jun et al. [8] discussed generalized derivations on ranked bigroupoids. They studied coderivations of ranked bigroupoids. In this paper, we discuss the generalization of coderivations of ranked bigroupoids. We introduce the notion of generalized coderivations in ranked bigroupoids. Combining a generalized self-coderivation with a rankomorphism, we obtain new generalized coderivations of ranked bigroupoids. From the notion of -derivation, we induce the existence of a rankomorphism of ranked bigroupoids.

#### 2. Preliminaries

Let be a nonempty set with a distinguished element . For any binary operationโโโโon , we consider the following axioms: โ(a1) ,โ(a2) ,โ(a3) ,โ(a4) and imply , โ(b1) , โ(b2) ,โ(b3) ,โ(b4) .

If satisfies axioms (a1), (a2), (a3), and (a4) under the binary operation , then we say that is a -algebra. If a -algebra satisfies the identity for all , we say that is a -*algebra*. Note that a -algebra satisfies conditions (b1), (b2), (b3), and (b4) under the binary operation (see [9]).

In a -semisimple -algebra , the following hold โ(b5) . โ(b6) .โ(b7) .โ(b8) implies . โ(b9) implies . โ(b10) implies . โ(b11) . โ(b12) .

A * ranked bigroupoid* (see [7]) is an algebraic system where is a nonempty set and โโ and โโ are binary operations defined on . We may consider the first binary operation as the major operation and the second binary operation as the minor operation.

Given a ranked bigroupoid , a map is called an -*self-derivation* (see [7]) if for all ,
In the same setting, a map is called an -*self-coderivation* (see [7]) if for all ,
Note that if is a commutative groupoid, then -self-derivations are -self-coderivations. A map is called an * abelian*--*self-derivation* (see [7]) if it is both an -self-derivation and an -self-coderivation.

Given ranked bigroupoids and , a map is called a * rankomorphism* (see [7]) if it satisfies and for all .

#### 3. Coderivations of Ranked Bigroupoids

*Definition 3.1 (see [8]). *Let be a ranked bigroupoid. A mapping is called a *generalized *-*self-derivation* if there exists an -self-derivation such that for all . If there exists an -self-coderivation such that for all , the mapping is called a *generalized *-*self-coderivation*. If is both a generalized -self-derivation and a generalized -self-coderivation, we say that is a *generalized abelian *-*self-derivation*.

*Definition 3.2 (see [7]). *Let and be ranked bigroupoids. A map is called an -*derivation* if there exists a rankomorphism such that
for all .

Theorem 3.3. *Let and be ranked bigroupoids with distinguished element in which the following items are valid.*(1)*The axioms (a3) and (b1) are valid under the major operation . *(2)*The axioms (b1), (b2), (b3), (a3) and (a4) are valid under the major operation . *(3)*The minor operation is defined by for all .**
If is an -derivation, then there exists a rankomorphism such that for all . In particular, .*

*Proof. *Assume that is an -derivation. Then there exists a rankomorphism such that
for all . Since the axiom (a3) is valid under the major operations and , we get . Hence
for all . It follows from (b3) that
for all . Note from (b2) and (a3) that
for all . Using (a4), we have for all . If we let , then .

Corollary 3.4. *Let and be ranked bigroupoids with distinguished element in which the following items are valid. *(1)*The axioms (a3) and (b1) are valid under the major operation . *(2)* is a BCI-algebra. *(3)*The minor operation is defined by for all .**
If is an -derivation, then there exists a rankomorphism such that for all . In particular, .*

Theorem 3.5. *Let and be ranked bigroupoids with distinguished element in which the following items are valid. *(1)*The axiom (a3) is valid under the major operation . *(2)*The axioms (a3), (b2), (b5), and (b6) are valid under the major operation .*(3)*The minor operation is defined by for all .**
If is an -derivation, then there exists a rankomorphism such that
*

*Proof. *Assume that is an -derivation. Then there exists a rankomorphism such that
for all . Let be such that . Then
and so
This completes the proof.

Corollary 3.6. *Let and be ranked bigroupoids with distinguished element in which the following items are valid. *(1)*The axiom (a3) is valid under the major operation . *(2)* is a -semisimple BCI-algebra. *(3)*The minor operation is defined by for all .**
If is an -derivation, then there exists a rankomorphism such that
*

*Definition 3.7 (see [8]). *Given ranked bigroupoids and , a map is called an *-coderivation* if there exists a rankomorphism such that
for all .

*Definition 3.8. *A map is called a *generalized *-*coderivation* if there exist both a rankomorphism and an -coderivation such that
for all .

Lemma 3.9 (see [8]). *If is a rankomorphism of ranked bigroupoids and is an -self-coderivation, then is an -coderivation. *

Theorem 3.10. *Let be a generalized -self-coderivation and a rankomorphism. Then is a generalized -coderivation. *

*Proof. *Since is a generalized -self-coderivation, there exists an -self-coderivation such that
for all . It follows that
for all . โNote from Lemma 3.9 that is an -coderivation. Hence is a generalized -coderivation.

Lemma 3.11 (see [8]). *If is a rankomorphism of ranked bigroupoids and is a -self-coderivation, then is an -coderivation. *

Theorem 3.12. *Let be a generalized -self-coderivation. If is a rankomorphism, then is a generalized -coderivation.*

*Proof. *Since is a generalized -self-coderivation, there exists a -self-coderivation such that
for all . It follows that
for all . By Lemma 3.11, is an -coderivation. Hence is a generalized -coderivation.

Lemma 3.13 (see [8]). *For ranked bigroupoids ,โโ and , consider a rankomorphism . If is a -coderivation, then is an -coderivation. *

Theorem 3.14. *Given ranked bigroupoids ,โโ, and , consider a rankomorphism . If is a generalized -coderivation, then is a generalized -coderivation. *

*Proof. *If is a generalized -coderivation, then there exist both a rankomorphism and a -coderivation such that
for all . It follows that
for all . Obviously, is a rankomorphism. By Lemma 3.13, is an -coderivation. Therefore is a generalized -coderivation.

Lemma 3.15 (see [8]). *For ranked bigroupoids ,โโ,โโand , consider a rankomorphism . If is an -coderivation, then is an -coderivation. *

Theorem 3.16. *Given ranked bigroupoids ,, and , consider a rankomorphism . If is a generalized -coderivation, then is a generalized -coderivation. *

*Proof. *If is a generalized -coderivation, then there exist both a rankomorphism and an -coderivation such that
for all . It follows that
for all . Obviously, is a rankomorphism and is an -coderivation by Lemma 3.15. This shows that is a generalized -coderivation.

#### Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions. Y. B. Jun, is an Executive Research Worker of Educational Research Institute Teachers College in Gyeongsang National University.

#### References

- H. E. Bell and L.-C. Kappe, โRings in which derivations satisfy certain algebraic conditions,โ
*Acta Mathematica Hungarica*, vol. 53, no. 3-4, pp. 339โ346, 1989. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H. E. Bell and G. Mason, โOn derivations in near-rings,โ in
*Near-Rings and Near-Dields*, vol. 137, pp. 31โ35, North-Holland, Amsterdam, The Netherlands, 1987. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - K. Kaya, โPrime rings with
*α*derivations,โ*Hacettepe Bulletin of Natural Sciences and Engineering*, vol. 16-17, pp. 63โ71, 1987-1988. View at Google Scholar - E. C. Posner, โDerivations in prime rings,โ
*Proceedings of the American Mathematical Society*, vol. 8, pp. 1093โ1100, 1957. View at Publisher ยท View at Google Scholar - Y. B. Jun and X. L. Xin, โOn derivations of BCI-algebras,โ
*Information Sciences*, vol. 159, no. 3-4, pp. 167โ176, 2004. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - N. O. Alshehri, โOn derivations of incline algebras,โ
*Scientiae Mathematicae Japonicae*, vol. 71, no. 3, pp. 349โ355, 2010. View at Google Scholar ยท View at Zentralblatt MATH - N. O. Alshehri, H. S. Kim, and J. Neggers, โDerivations on ranked bigroupoids,โ
*Applied Mathematics & Information Sciences*. In press. - Y. B. Jun, E. H. Roh, and G. Muhiuddin, โGeneralized derivations on ranked bi-groupoids,โ
*Knowledge-Based Systems*. Under review process. - Y. Huang,
*BCI-Algebra*, Science Press, Beijing, China, 2006.