Journal of Applied Mathematics
Volume 2012, Article ID 783657, 9 pages
http://dx.doi.org/10.1155/2012/783657
Research Article

Further Results on Derivations of Ranked Bigroupoids

1Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea
2Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
3Department of Mathematics Education, Chinju National University of Education, Jinju 660-756, Republic of Korea

Received 16 May 2012; Revised 2 August 2012; Accepted 4 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

Further properties on -self-(co)derivations of ranked bigroupoids are investigated, and conditions for an -self-(co)derivation to be regular are provided. The notion of ranked -subsystems is introduced, and related properties are investigated.

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. Zhan and Liu  studied -derivations 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. In this paper, we investigate further properties on -self-(co)derivations and provide conditions for an -self-(co)derivation to be regular. We introduce the notion of ranked -subsystems and investigate related properties.

2. Preliminaries

In a nonempty set with a constant and a binary operation , we consider the following axioms: (a1)(a2)(a3)(a4), (b1), (b2)(b3), (b4).

If satisfies axioms (a1), (a2), (a3), and (a4), then we say that is a -algebra. Note that a -algebra satisfies conditions (b1), (b2), (b3), and (b4) (see ).

In a -semisimple -algebra , the following hold: (b5), (b6).

3. Derivations on Ranked Bigroupoids

A ranked bigroupoid (see ) is an algebraic system where is a non-empty 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 -self-derivation (see ) if it is both an -self-derivation and an -self-coderivation.

Proposition 3.1. Let be a ranked bigroupoid with distinguished element in which the minor operation is defined by for all .(1)Assume that satisfies axioms (b1), (b2), (b3), (a3), and (a4). If a map is an -self-derivation, then for all .(2)If satisfies two axioms (b1) and (a3) and a map is an -self-coderivation, then the following are equivalent:(2.1) ;(2.2) .

Proof. (1) Let . Using (b1) and (b2), we have It follows from (b3) that Using (b2) and (a3), we have , and so for all by (a4).
(2) Let be an -self-coderivation. If , then for all . Assume that for all . Taking implies that .

Corollary 3.2. Let be a ranked bigroupoid in which is a -algebra and the minor operation is defined by for all .(1)If a map is an -self-derivation, then for all .(2)If a map is an -self-coderivation, then the following are equivalent:  (2.1);  (2.2).

Lemma 3.3. Let be a ranked bigroupoid with distinguished element in which three axioms ,, and are valid and the minor operation is defined by for all .(1)For every with , one has (2)If an element of satisfies , then for all .

Proof. (1) Let be such that . Then and so by (a4).
(2) Since , it follows from (3.6) that for all .

Corollary 3.4. Let be a ranked bigroupoid in which is a -algebra and the minor operation is defined by for all .(1)For every with , one has (2)If an element of satisfies , then for all .

Proposition 3.5. Let be a ranked bigroupoid with distinguished element in which four axioms ,, , and are valid and the minor operation is defined by for all . If a map is an -self-coderivation, then for all with .

Proof. Let be such that . Since , it follows from Lemma 3.3(2) that .

Corollary 3.6. Let be a ranked bigroupoid in which is a -algebra and the minor operation is defined by for all . If a map is an -self-coderivation, then for all with .

Using Proposition 3.5, we can find an -self-derivation which is not an -self-coderivation.

Example 3.7. Let be a ranked bigroupoid where is the set of all integers with the minus operation “−” and the minor operation “” defined by for all . Let be a self map of given by for all . Then is a -self-derivation since Note that . Hence is not a -self-coderivation by Proposition 3.5.

Proposition 3.8. Let be a ranked bigroupoid with distinguished element and the minor operation is defined by for all . For an -self-derivation , if satisfies axioms ,, and , then for all . Moreover, if , then is an identity map.

Proof. Assume that satisfies axioms (b2), (b5), and (b6). Then for all . Moreover, if then for all , and so is an identity map.

Corollary 3.9. Let be a ranked bigroupoid in which is a -algebra and the minor operation is defined by for all . If a map is an -self-derivation, then(1); (2)if is -semisimple, then for all ;(3)if is -semisimple and , then is an identity map.

Definition 3.10. Let be a ranked bigroupoid with distinguished element . A self map of is said to be regular if .

Example 3.11. Consider a ranked bigroupoid in which and binary operations “” and “” are defined by Define a map by Then is an abelian -self-derivation which is regular.

Proposition 3.12. Let be a ranked bigroupoid with distinguished element in which the minor operation is defined by for all and for all . Then every -self-derivation is regular. Moreover, if satisfies the axioms and then every -self-coderivation is regular.

Proof. Let be an -self-derivation. Then If is an -self-coderivation, then Hence every -self-(co)derivation is regular.

Proposition 3.13. Let be a ranked bigroupoid with distinguished element in which the minor operation is defined by for all and two axioms and are satisfied. Let be a self map of and such that (resp., ) for all . If is an -self-derivation (resp., -self-coderivation), then it is regular.

Proof. Assume that is an -self-derivation. For any , we have which implies that Hence is regular. Now, let be an -self-coderivation such that for all . Then and so Therefore is regular.

Definition 3.14. Let be a ranked bigroupoid with distinguished element . Let be a self map of . A subset of is called a ranked -subsystem of if it satisfies the following:(r1), (r2).
Moreover, if a ranked -subsystem of satisfies , then we say that is ranked -invariant.

Example 3.15. Consider a ranked bigroupoid in which and binary operations “”and “” are defined by and for all . Define a map d: by Then is an abelian -self-derivation which is not regular. It is easily check that is a ranked -subsystem of . Since , is not ranked -invariant.

Example 3.16. In Example 3.11, is a ranked -invariant -subsystem of .

Theorem 3.17. Let be a ranked bigroupoid with distinguished element in which three axioms ,, and are valid and the minor operation is defined by for all . For an -self-coderivation , if is regular then every ranked -subsystem of is ranked -invariant.

Proof. Assume that is regular and let be a ranked -subsystem of . Then for all by Proposition 3.1(2). Let . Then for some . Thus , and so by (r2). Hence and is ranked -invariant.

Corollary 3.18. Let be an -self-coderivation where is a -algebra and the minor operation is defined by for all . If is regular, then every ideal of is ranked -invariant.

Example 3.15 shows that Theorem 3.17 is not true if we drop the regularity of .

We consider the converse of Theorem 3.17.

Theorem 3.19. Let be an -self-coderivation where is a ranked bigroupoid with distinguished element in which the minor operation is defined by for all and there does not exist a nonzero element of such that . If every ranked -subsystem of is ranked -invariant, then is regular.

Proof. Assume that every ranked -subsystem of is ranked -invariant. Note that is a ranked -subsystem of . Thus , and therefore is regular.

Corollary 3.20. Let be an -self-coderivation where is a -algebra and the minor operation is defined by for all . Then is regular if and only if every ranked -subsystem of is ranked -invariant.

Proposition 3.21. Let be a ranked bigroupoid where is a -algebra and the minor operation is defined by for all . For any , let be a self map of defined by for all . If satisfies the following conditions:(1) for all ,(2), then is an abelian -self-derivation.

Proof. If satisfies two given conditions, then the following identity is valid (see ): It follows from (b1), (a3), and (b2) that Hence is an -self-derivation. Similarly, we can verify that is an -self-coderivation.

Corollary 3.22. Let be a ranked bigroupoid where is a -algebra and the minor operation is defined by for all . For any , let be a self map of defined by for all . If satisfies and the following conditions: (1) for all ,(2) for all , then is an abelian -self-derivation.

Proof. If satisfies both (b1) and the second condition, then is a -semisimple -algebra (see ). Hence the second condition of Proposition 3.21 is valid. Therefore is an abelian -self-derivation.

4. Conclusion

Alshehri et al.  introduced the notion of ranked bigroupoids and discussed -self-(co)derivations.

A nonempty set together with maps and is called a ranked bigroupoid. For a ranked bigroupoid , a map is called: (1)an -self-derivation if for all ;(2)an -self-coderivation if for all .

In this paper, we have investigated further properties on -self-(co)derivations and have provided conditions for an -self-(co)derivation to be regular. We have introduced the notion of ranked -subsystems and have investigated related properties.

In general, there are many kind of derivations (generalized derivations, biderivations, triderivations, etc.) in algebraic structures, for example, (near) rings, prime rings, semiprime rings, -near-rings, incline algebras, Banach algebras, lattices, MV-algebras, and BCK/BCI-algebras.

Based on this paper together with related papers on derivations, we will consider several kind of derivations in ranked bigroupoids.

Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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