Journal of Applied Mathematics
Volume 2012, Article ID 626781, 8 pages
http://dx.doi.org/10.1155/2012/626781
Research Article

Coderivations of Ranked Bigroupoids

1Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
2Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea
3School 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-Keung Lam

Copyright © 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  have studied derivations in rings and near-rings. Jun and Xin  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  applied the notion of derivations to incline algebras. Alshehri et al.  introduced the notion of ranked bigroupoids and discussed -self-(co)derivations. Jun et al.  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 operation    on , we consider the following axioms: (a1) ,(a2) ,(a3) ,(a4) and imply , (b1) , (b2) ,(b3) ,(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 ).

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

A ranked bigroupoid (see ) is an algebraic system where is a nonempty set and “” and “” 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 ) if for all , In the same setting, a map is called an -self-coderivation (see ) 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 ) if it is both an -self-derivation and an -self-coderivation.

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

3. Coderivations of Ranked Bigroupoids

Definition 3.1 (see ). 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 ). 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 ). 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 ). 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 .  Note from Lemma 3.9 that is an -coderivation. Hence is a generalized -coderivation.

Lemma 3.11 (see ). 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 ). For ranked bigroupoids ,   and , consider a rankomorphism . If is a -coderivation, then is an -coderivation.

Theorem 3.14. 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 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 ). For ranked bigroupoids ,  ,  and , 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.

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