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Journal of Applied Mathematics
Volumeย 2012, Article IDย 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-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 (๐‘‹,โˆ—,&)-derivation, the existence of a rankomorphism of ranked bigroupoids is established.

1. Introduction

Several authors [1โ€“4] have studied derivations in rings and near-rings. Jun and Xin [5] 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. Alshehri [6] applied the notion of derivations to incline algebras. Alshehri et al. [7] introduced the notion of ranked bigroupoids and discussed (๐‘‹,โˆ—,&)-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 (๐‘‹,โˆ—,&)-derivation, we induce the existence of a rankomorphism of ranked bigroupoids.

2. Preliminaries

Let ๐‘‹ be a nonempty set with a distinguished element 0. For any binary operationโ€‰โ€‰โ™ฎโ€‰โ€‰on ๐‘‹, we consider the following axioms: โ€‰(a1) ((๐‘ฅโ™ฎ๐‘ฆ)โ™ฎ(๐‘ฅโ™ฎ๐‘ง))โ™ฎ(๐‘งโ™ฎ๐‘ฆ)=0,โ€‰(a2) (๐‘ฅโ™ฎ(๐‘ฅโ™ฎ๐‘ฆ))โ™ฎ๐‘ฆ=0,โ€‰(a3) ๐‘ฅโ™ฎ๐‘ฅ=0,โ€‰(a4) ๐‘ฅโ™ฎ๐‘ฆ=0 and ๐‘ฆโ™ฎ๐‘ฅ=0 imply ๐‘ฅ=๐‘ฆ, โ€‰(b1) ๐‘ฅโ™ฎ0=๐‘ฅ, โ€‰(b2) (๐‘ฅโ™ฎ๐‘ฆ)โ™ฎ๐‘ง=(๐‘ฅโ™ฎ๐‘ง)โ™ฎ๐‘ฆ,โ€‰(b3) ((๐‘ฅโ™ฎ๐‘ง)โ™ฎ(๐‘ฆโ™ฎ๐‘ง))โ™ฎ(๐‘ฅโ™ฎ๐‘ฆ)=0,โ€‰(b4) ๐‘ฅโ™ฎ(๐‘ฅโ™ฎ(๐‘ฅโ™ฎ๐‘ฆ))=๐‘ฅโ™ฎ๐‘ฆ.

If ๐‘‹ satisfies axioms (a1), (a2), (a3), and (a4) under the binary operation โˆ—, then we say that (๐‘‹,โˆ—,0) is a BCI-algebra. If a BCI-algebra (๐‘‹,โˆ—,0) satisfies the identity 0โˆ—๐‘ฅ=0 for all ๐‘ฅโˆˆ๐‘‹, we say that (๐‘‹,โˆ—,0) is a BCK-algebra. Note that a BCI-algebra (๐‘‹,โˆ—,0) satisfies conditions (b1), (b2), (b3), and (b4) under the binary operation โˆ— (see [9]).

In a ๐‘-semisimple BCI-algebra (๐‘‹,โˆ—,0), the following hold โ€‰(b5) (๐‘ฅโˆ—๐‘ง)โˆ—(๐‘ฆโˆ—๐‘ง)=๐‘ฅโˆ—๐‘ฆ. โ€‰(b6) 0โˆ—(0โˆ—๐‘ฅ)=๐‘ฅ.โ€‰(b7) ๐‘ฅโˆ—(0โˆ—๐‘ฆ)=๐‘ฆโˆ—(0โˆ—๐‘ฅ).โ€‰(b8) ๐‘ฅโˆ—๐‘ฆ=0 implies ๐‘ฅ=๐‘ฆ. โ€‰(b9) ๐‘ฅโˆ—๐‘Ž=๐‘ฅโˆ—๐‘ implies ๐‘Ž=๐‘. โ€‰(b10) ๐‘Žโˆ—๐‘ฅ=๐‘โˆ—๐‘ฅ implies ๐‘Ž=๐‘. โ€‰(b11) ๐‘Žโˆ—(๐‘Žโˆ—๐‘ฅ)=๐‘ฅ. โ€‰(b12) ((๐‘ฅโˆ—๐‘ฆ)โˆ—๐‘ง)โˆ—(๐‘ฅโˆ—(๐‘ฆโˆ—๐‘ง))=0.

A ranked bigroupoid (see [7]) is an algebraic system (๐‘‹,โˆ—,โ€ข) where ๐‘‹ is a nonempty set and โ€œโˆ—โ€ and โ€œโ€ขโ€ are binary operations defined on ๐‘‹. We may consider the first binary operation โˆ— as the major operation and the second binary operation โ€ข as the minor operation.

Given a ranked bigroupoid (๐‘‹,โˆ—,&), a map ๐‘‘โˆถ๐‘‹โ†’๐‘‹ is called an (๐‘‹,โˆ—,&)-self-derivation (see [7]) if for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, ๐‘‘(๐‘ฅโˆ—๐‘ฆ)=(๐‘‘(๐‘ฅ)โˆ—๐‘ฆ)&(๐‘ฅโˆ—๐‘‘(๐‘ฆ)).(2.1) In the same setting, a map ๐‘‘โˆถ๐‘‹โ†’๐‘‹ is called an (๐‘‹,โˆ—,&)-self-coderivation (see [7]) if for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, ๐‘‘(๐‘ฅโˆ—๐‘ฆ)=(๐‘ฅโˆ—๐‘‘(๐‘ฆ))&(๐‘‘(๐‘ฅ)โˆ—๐‘ฆ).(2.2) Note that if (๐‘‹,โˆ—) is a commutative groupoid, then (๐‘‹,โˆ—,&)-self-derivations are (๐‘‹,โˆ—,&)-self-coderivations. A map ๐‘‘โˆถ๐‘‹โ†’๐‘‹ is called an abelian-(๐‘‹,โˆ—,&)-self-derivation (see [7]) if it is both an (๐‘‹,โˆ—,&)-self-derivation and an (๐‘‹,โˆ—,&)-self-coderivation.

Given ranked bigroupoids (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”), a map ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is called a rankomorphism (see [7]) if it satisfies ๐‘“(๐‘ฅโˆ—๐‘ฆ)=๐‘“(๐‘ฅ)โ€ข๐‘“(๐‘ฆ) and ๐‘“(๐‘ฅ&๐‘ฆ)=๐‘“(๐‘ฅ)๐œ”๐‘“(๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹.

3. Coderivations of Ranked Bigroupoids

Definition 3.1 (see [8]). Let (๐‘‹,โˆ—,&) be a ranked bigroupoid. A mapping ๐ทโˆถ๐‘‹โ†’๐‘‹ is called a generalized (๐‘‹,โˆ—,&)-self-derivation if there exists an (๐‘‹,โˆ—,&)-self-derivation ๐‘‘โˆถ๐‘‹โ†’๐‘‹ such that ๐ท(๐‘ฅโˆ—๐‘ฆ)=(๐ท(๐‘ฅ)โˆ—๐‘ฆ)&(๐‘ฅโˆ—๐‘‘(๐‘ฆ)) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. If there exists an (๐‘‹,โˆ—,&)-self-coderivation ๐‘‘โˆถ๐‘‹โ†’๐‘‹ such that ๐ท(๐‘ฅโˆ—๐‘ฆ)=(๐‘ฅโˆ—๐ท(๐‘ฆ))&(๐‘‘(๐‘ฅ)โˆ—๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, the mapping ๐ทโˆถ๐‘‹โ†’๐‘‹ is called a generalized (๐‘‹,โˆ—,&)-self-coderivation. If ๐ท is both a generalized (๐‘‹,โˆ—,&)-self-derivation and a generalized (๐‘‹,โˆ—,&)-self-coderivation, we say that ๐ท is a generalized abelian (๐‘‹,โˆ—,&)-self-derivation.

Definition 3.2 (see [7]). Let (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”) be ranked bigroupoids. A map ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is called an (๐‘‹,โˆ—,&)-derivation if there exists a rankomorphism ๐‘“โˆถ๐‘‹โ†’๐‘Œ such that ๐›ฟ(๐‘ฅโˆ—๐‘ฆ)=(๐›ฟ(๐‘ฅ)โ€ข๐‘“(๐‘ฆ))๐œ”(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฆ)),(3.1) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹.

Theorem 3.3. Let (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”) be ranked bigroupoids with distinguished element 0 in which the following items are valid.(1)The axioms (a3) and (b1) are valid under the major operation โˆ—. (2)The axioms (b1), (b2), (b3), (a3) and (a4) are valid under the major operation โ€ข. (3)The minor operation ๐œ” is defined by ๐‘Ž๐œ”๐‘=๐‘โ€ข(๐‘โ€ข๐‘Ž) for all ๐‘Ž,๐‘โˆˆ๐‘Œ.
If ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-derivation, then there exists a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that ๐›ฟ(๐‘ฅ)=๐›ฟ(๐‘ฅ)๐œ”๐‘“(๐‘ฅ) for all ๐‘ฅโˆˆ๐‘‹. In particular, ๐›ฟ(0)=0โ€ข(0โ€ข๐›ฟ(0)).

Proof. Assume that ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-derivation. Then there exists a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that ๐›ฟ(๐‘ฅโˆ—๐‘ฆ)=(๐›ฟ(๐‘ฅ)โ€ข๐‘“(๐‘ฆ))๐œ”(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฆ)),(3.2) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Since the axiom (a3) is valid under the major operations โˆ— and โ€ข, we get ๐‘“(0)=0. Hence =๐›ฟ(๐‘ฅ)=๐›ฟ(๐‘ฅโˆ—0)=(๐›ฟ(๐‘ฅ)โ€ข๐‘“(0))๐œ”(๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))(๐›ฟ(๐‘ฅ)โ€ข0)๐œ”(๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))=๐›ฟ(๐‘ฅ)๐œ”(๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))=(๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))โ€ข((๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))โ€ข๐›ฟ(๐‘ฅ))=(๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))โ€ข((๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ))โ€ข๐›ฟ(0)),(3.3) for all ๐‘ฅโˆˆ๐‘‹. It follows from (b3) that =๐›ฟ(๐‘ฅ)โ€ข(๐›ฟ(๐‘ฅ)๐œ”๐‘“(๐‘ฅ))=๐›ฟ(๐‘ฅ)โ€ข(๐‘“(๐‘ฅ)โ€ข(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ)))((๐‘“(๐‘ฅ)โ€ข๐›ฟ(0))โ€ข((๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ))โ€ข๐›ฟ(0)))โ€ข(๐‘“(๐‘ฅ)โ€ข(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ)))=0,(3.4) for all ๐‘ฅโˆˆ๐‘‹. Note from (b2) and (a3) that =(๐›ฟ(๐‘ฅ)๐œ”๐‘“(๐‘ฅ))โ€ข๐›ฟ(๐‘ฅ)=(๐‘“(๐‘ฅ)โ€ข(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ)))โ€ข๐›ฟ(๐‘ฅ)(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ))โ€ข(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฅ))=0,(3.5) for all ๐‘ฅโˆˆ๐‘‹. Using (a4), we have ๐›ฟ(๐‘ฅ)=๐›ฟ(๐‘ฅ)๐œ”๐‘“(๐‘ฅ) for all ๐‘ฅโˆˆ๐‘‹. If we let ๐‘ฅโˆถ=0, then ๐›ฟ(0)=0โ€ข(0โ€ข๐›ฟ(0)).

Corollary 3.4. Let (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”) be ranked bigroupoids with distinguished element 0 in which the following items are valid. (1)The axioms (a3) and (b1) are valid under the major operation โˆ—. (2)(๐‘Œ,โ€ข,0) is a BCI-algebra. (3)The minor operation ๐œ” is defined by ๐‘Ž๐œ”๐‘=๐‘โ€ข(๐‘โ€ข๐‘Ž) for all ๐‘Ž,๐‘โˆˆ๐‘Œ.
If ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-derivation, then there exists a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that ๐›ฟ(๐‘ฅ)=๐›ฟ(๐‘ฅ)๐œ”๐‘“(๐‘ฅ) for all ๐‘ฅโˆˆ๐‘‹. In particular, ๐›ฟ(0)=0โ€ข(0โ€ข๐›ฟ(0)).

Theorem 3.5. Let (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”) be ranked bigroupoids with distinguished element 0 in which the following items are valid. (1)The axiom (a3) is valid under the major operation โˆ—. (2)The axioms (a3), (b2), (b5), and (b6) are valid under the major operation โ€ข.(3)The minor operation ๐œ” is defined by ๐‘Ž๐œ”๐‘=๐‘โ€ข(๐‘โ€ข๐‘Ž) for all ๐‘Ž,๐‘โˆˆ๐‘Œ.
If ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-derivation, then there exists a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that (โˆ€๐‘Žโˆˆ๐‘‹)(๐‘Ž=0โˆ—(0โˆ—๐‘Ž)โŸน๐›ฟ(๐‘Ž)=๐›ฟ(0)โ€ข(0โ€ข๐‘“(๐‘Ž))).(3.6)

Proof. Assume that ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-derivation. Then there exists a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that ๐›ฟ(๐‘ฅโˆ—๐‘ฆ)=(๐›ฟ(๐‘ฅ)โ€ข๐‘“(๐‘ฆ))๐œ”(๐‘“(๐‘ฅ)โ€ข๐›ฟ(๐‘ฆ))(3.7) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Let ๐‘Žโˆˆ๐‘‹ be such that ๐‘Ž=0โˆ—(0โˆ—๐‘Ž). Then ๐‘“(๐‘Ž)=๐‘“(0โˆ—(0โˆ—๐‘Ž))=๐‘“(0)โ€ข(๐‘“(0)โ€ข๐‘“(๐‘Ž))=0โ€ข(0โ€ข๐‘“(๐‘Ž)),(3.8) and so =๐›ฟ(๐‘Ž)=๐›ฟ(0โˆ—(0โˆ—๐‘Ž))=(๐›ฟ(0)โ€ข๐‘“(0โˆ—๐‘Ž))๐œ”(๐‘“(0)โ€ข๐›ฟ(0โˆ—๐‘Ž))(๐›ฟ(0)โ€ข๐‘“(0โˆ—๐‘Ž))๐œ”(0โ€ข๐›ฟ(0โˆ—๐‘Ž))=(0โ€ข๐›ฟ(0โˆ—๐‘Ž))โ€ข((0โ€ข๐›ฟ(0โˆ—๐‘Ž))โ€ข(๐›ฟ(0)โ€ข๐‘“(0โˆ—๐‘Ž)))=(0โ€ข๐›ฟ(0โˆ—๐‘Ž))โ€ข((0โ€ข(๐›ฟ(0)โ€ข๐‘“(0โˆ—๐‘Ž)))โ€ข๐›ฟ(0โˆ—๐‘Ž))=0โ€ข(0โ€ข(๐›ฟ(0)โ€ข๐‘“(0โˆ—๐‘Ž)))=0โ€ข(0โ€ข(๐›ฟ(0)โ€ข(๐‘“(0)โ€ข๐‘“(๐‘Ž))))=0โ€ข(0โ€ข(๐›ฟ(0)โ€ข(0โ€ข๐‘“(๐‘Ž))))=๐›ฟ(0)โ€ข(0โ€ข๐‘“(๐‘Ž)).(3.9) This completes the proof.

Corollary 3.6. Let (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”) be ranked bigroupoids with distinguished element 0 in which the following items are valid. (1)The axiom (a3) is valid under the major operation โˆ—. (2)(๐‘Œ,โ€ข,0) is a ๐‘-semisimple BCI-algebra. (3)The minor operation ๐œ” is defined by ๐‘Ž๐œ”๐‘=๐‘โ€ข(๐‘โ€ข๐‘Ž) for all ๐‘Ž,๐‘โˆˆ๐‘Œ.
If ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-derivation, then there exists a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that (โˆ€๐‘Žโˆˆ๐‘‹)(๐‘Ž=0โˆ—(0โˆ—๐‘Ž)โŸน๐›ฟ(๐‘Ž)=๐›ฟ(0)โ€ข(0โ€ข๐‘“(๐‘Ž))).(3.10)

Definition 3.7 (see [8]). Given ranked bigroupoids (๐‘‹,โˆ—,&) and (๐‘Œ,โ€ข,๐œ”), a map ๐œ‘โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is called an (๐‘‹,โˆ—,&)-coderivation if there exists a rankomorphism ๐‘“โˆถ๐‘‹โ†’๐‘Œ such that ๐œ‘(๐‘ฅโˆ—๐‘ฆ)=(๐‘“(๐‘ฅ)โ€ข๐œ‘(๐‘ฆ))๐œ”(๐œ‘(๐‘ฅ)โ€ข๐‘“(๐‘ฆ)),(3.11) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹.

Definition 3.8. A map ฮจโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is called a generalized (๐‘‹,โˆ—,&)-coderivation if there exist both a rankomorphism ๐‘“โˆถ๐‘‹โ†’๐‘Œ and an (๐‘‹,โˆ—,&)-coderivation ๐›ฟโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that ฮจ(๐‘ฅโˆ—๐‘ฆ)=(๐‘“(๐‘ฅ)โ€ขฮจ(๐‘ฆ))๐œ”(๐›ฟ(๐‘ฅ)โ€ข๐‘“(๐‘ฆ)),(3.12) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹.

Lemma 3.9 (see [8]). If ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a rankomorphism of ranked bigroupoids and ๐‘‘โˆถ๐‘‹โ†’๐‘‹ is an (๐‘‹,โˆ—,&)-self-coderivation, then ๐‘“โˆ˜๐‘‘โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-coderivation.

Theorem 3.10. Let ๐ทโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘‹,โˆ—,&) be a generalized (๐‘‹,โˆ—,&)-self-coderivation and ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) a rankomorphism. Then ๐‘“โˆ˜๐ท is a generalized (๐‘‹,โˆ—,&)-coderivation.

Proof. Since ๐ท is a generalized (๐‘‹,โˆ—,&)-self-coderivation, there exists an (๐‘‹,โˆ—,&)-self-coderivation ๐‘‘โˆถ๐‘‹โ†’๐‘‹ such that ๐ท(๐‘ฅโˆ—๐‘ฆ)=(๐‘ฅโˆ—๐ท(๐‘ฆ))&(๐‘‘(๐‘ฅ)โˆ—๐‘ฆ),(3.13) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. It follows that (๐‘“โˆ˜๐ท)(๐‘ฅโˆ—๐‘ฆ)=๐‘“(๐ท(๐‘ฅโˆ—๐‘ฆ))=๐‘“((๐‘ฅโˆ—๐ท(๐‘ฆ))&(๐‘‘(๐‘ฅ)โˆ—๐‘ฆ))=๐‘“(๐‘ฅโˆ—๐ท(๐‘ฆ))๐œ”๐‘“(๐‘‘(๐‘ฅ)โˆ—๐‘ฆ)=(๐‘“(๐‘ฅ)โ€ข(๐‘“โˆ˜๐ท)(๐‘ฆ))๐œ”((๐‘“โˆ˜๐‘‘)(๐‘ฅ)โ€ข๐‘“(๐‘ฆ)),(3.14) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. โ€‰Note from Lemma 3.9 that ๐‘“โˆ˜๐‘‘ is an (๐‘‹,โˆ—,&)-coderivation. Hence ๐‘“โˆ˜๐ท is a generalized (๐‘‹,โˆ—,&)-coderivation.

Lemma 3.11 (see [8]). If ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a rankomorphism of ranked bigroupoids and ๐›ฟโˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘Œ,โ€ข,๐œ”) is a (๐‘Œ,โ€ข,๐œ”)-self-coderivation, then ๐›ฟโˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-coderivation.

Theorem 3.12. Let ๐ทโˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘Œ,โ€ข,๐œ”) be a generalized (๐‘Œ,โ€ข,๐œ”)-self-coderivation. If ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a rankomorphism, then ๐ทโˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a generalized (๐‘‹,โˆ—,&)-coderivation.

Proof. Since ๐ท is a generalized (๐‘Œ,โ€ข,๐œ”)-self-coderivation, there exists a (๐‘Œ,โ€ข,๐œ”)-self-coderivation ๐›ฟโˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘‹,โˆ—,&) such that ๐ท(๐‘Žโ€ข๐‘)=(๐‘Žโ€ข๐ท(๐‘))๐œ”(๐›ฟ(๐‘Ž)โ€ข๐‘),(3.15) for all ๐‘Ž,๐‘โˆˆ๐‘Œ. It follows that =(๐ทโˆ˜๐‘“)(๐‘ฅโˆ—๐‘ฆ)=๐ท(๐‘“(๐‘ฅโˆ—๐‘ฆ))=๐ท(๐‘“(๐‘ฅ)โ€ข๐‘“(๐‘ฆ))(๐‘“(๐‘ฅ)โ€ข๐ท(๐‘“(๐‘ฆ)))๐œ”(๐›ฟ(๐‘“(๐‘ฅ))โ€ข๐‘“(๐‘ฆ))=(๐‘“(๐‘ฅ)โ€ข(๐ทโˆ˜๐‘“)(๐‘ฆ))๐œ”((๐›ฟโˆ˜๐‘“)(๐‘ฅ)โ€ข๐‘“(๐‘ฆ)),(3.16) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. By Lemma 3.11, ๐›ฟโˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-coderivation. Hence ๐ทโˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a generalized (๐‘‹,โˆ—,&)-coderivation.

Lemma 3.13 (see [8]). For ranked bigroupoids (๐‘‹,โˆ—,&),โ€‰โ€‰(๐‘Œ,โ€ข,๐œ”) and (๐‘,โ–ก,๐œ‹), consider a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”). If ๐œ‘โˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹) is a (๐‘Œ,โ€ข,๐œ”)-coderivation, then ๐œ‘โˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘,โ–ก,๐œ‹) is an (๐‘‹,โˆ—,&)-coderivation.

Theorem 3.14. Given ranked bigroupoids (๐‘‹,โˆ—,&),โ€‰โ€‰(๐‘Œ,โ€ข,๐œ”), and (๐‘,โ–ก,๐œ‹), consider a rankomorphism ๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”). If ฮจโˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹) is a generalized (๐‘Œ,โ€ข,๐œ”)-coderivation, then ฮจโˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘,โ–ก,๐œ‹) is a generalized (๐‘‹,โˆ—,&)-coderivation.

Proof. If ฮจโˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹) is a generalized (๐‘Œ,โ€ข,๐œ”)-coderivation, then there exist both a rankomorphism ๐‘”โˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹) and a (๐‘Œ,โ€ข,๐œ”)-coderivation ๐œ‘โˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹) such that ฮจ(๐‘Žโ€ข๐‘)=(๐‘”(๐‘Ž)โ–กฮจ(๐‘))๐œ‹(๐œ‘(๐‘Ž)โ–ก๐‘”(๐‘)),(3.17) for all ๐‘Ž,๐‘โˆˆ๐‘Œ. It follows that =(ฮจโˆ˜๐‘“)(๐‘ฅโˆ—๐‘ฆ)=ฮจ(๐‘“(๐‘ฅโˆ—๐‘ฆ))=ฮจ(๐‘“(๐‘ฅ)โ€ข๐‘“(๐‘ฆ))(๐‘”(๐‘“(๐‘ฆ))๐œ”ฮจ(๐‘“(๐‘ฆ)))๐œ‹(๐œ‘(๐‘“(๐‘ฅ))โ–ก๐‘”(๐‘“(๐‘ฆ)))=((๐‘”โˆ˜๐‘“)(๐‘ฅ)โ–ก(๐œ‘โˆ˜๐‘“)(๐‘ฆ))๐œ‹((๐œ‘โˆ˜๐‘“)(๐‘ฅ)โ–ก(๐‘”โˆ˜๐‘“)(๐‘ฆ)),(3.18) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Obviously, ๐‘”โˆ˜๐‘“ is a rankomorphism. By Lemma 3.13, ๐œ‘โˆ˜๐‘“ is an (๐‘‹,โˆ—,&)-coderivation. Therefore ฮจโˆ˜๐‘“โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘,โ–ก,๐œ‹) is a generalized (๐‘‹,โˆ—,&)-coderivation.

Lemma 3.15 (see [8]). For ranked bigroupoids (๐‘‹,โˆ—,&),โ€‰โ€‰(๐‘Œ,โ€ข,๐œ”),โ€‰โ€‰and (๐‘,โ–ก,๐œ‹), consider a rankomorphism ๐‘”โˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹). If ๐œ‚โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is an (๐‘‹,โˆ—,&)-coderivation, then ๐‘”โˆ˜๐œ‚โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘,โ–ก,๐œ‹) is an (๐‘‹,โˆ—,&)-coderivation.

Theorem 3.16. Given ranked bigroupoids (๐‘‹,โˆ—,&),(๐‘Œ,โ€ข,๐œ”), and (๐‘,โ–ก,๐œ‹), consider a rankomorphism ๐‘”โˆถ(๐‘Œ,โ€ข,๐œ”)โ†’(๐‘,โ–ก,๐œ‹). If ฮฆโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a generalized (๐‘‹,โˆ—,&)-coderivation, then ๐‘”โˆ˜ฮฆโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘,โ–ก,๐œ‹) is a generalized (๐‘‹,โˆ—,&)-coderivation.

Proof. If ฮฆโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) is a generalized (๐‘‹,โˆ—,&)-coderivation, then there exist both a rankomorphism ๐‘“โˆถ๐‘‹โ†’๐‘Œ and an (๐‘‹,โˆ—,&)-coderivation ๐œ‚โˆถ(๐‘‹,โˆ—,&)โ†’(๐‘Œ,โ€ข,๐œ”) such that ฮฆ(๐‘ฅโˆ—๐‘ฆ)=(๐‘“(๐‘ฅ)โ€ขฮฆ(๐‘ฆ))๐œ”(๐œ‚(๐‘ฅ)โ€ข๐‘“(๐‘ฆ)),(3.19) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. It follows that (๐‘”โˆ˜ฮฆ)(๐‘ฅโˆ—๐‘ฆ)=๐‘”(ฮฆ(๐‘ฅโˆ—๐‘ฆ))=๐‘”((๐‘“(๐‘ฅ)โ€ขฮฆ(๐‘ฆ))๐œ”(๐œ‚(๐‘ฅ)โ€ข๐‘“(๐‘ฆ)))=๐‘”(๐‘“(๐‘ฅ)โ€ขฮฆ(๐‘ฆ))๐œ‹๐‘”(๐œ‚(๐‘ฅ)โ€ข๐‘“(๐‘ฆ))=(๐‘”(๐‘“(๐‘ฅ))โ–ก๐‘”(ฮฆ(๐‘ฆ)))๐œ‹(๐‘”(๐œ‚(๐‘ฅ))โ–ก๐‘”(๐‘“(๐‘ฆ)))=((๐‘”โˆ˜๐‘“)(๐‘ฅ)โ–ก(๐‘”โˆ˜ฮฆ)(๐‘ฆ))๐œ‹((๐‘”โˆ˜๐œ‚)(๐‘ฅ)โ–ก(๐‘”โˆ˜๐‘“)(๐‘ฆ)),(3.20) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Obviously, ๐‘”โˆ˜๐‘“ is a rankomorphism and ๐‘”โˆ˜๐œ‚ is an (๐‘‹,โˆ—,&)-coderivation by Lemma 3.15. This shows that ๐‘”โˆ˜ฮฆโˆถ(๐‘‹,โˆ—,&)โ†’(๐‘,โ–ก,๐œ‹) is a generalized (๐‘‹,โˆ—,&)-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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