# Further Results on Derivations of  Ranked Bigroupoids

Authors: Young Bae Jun; Hee Sik Kim; Eun Hwan Roh
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

Several authors [1–4] 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.  Zhan and Liu  [6] studied f-derivations in BCI-algebras. Alshehri [7] applied the notion of derivations to incline algebras. Alshehri et al. [8] introduced the notion of ranked bigroupoids and discussed (X,*,&)-self-(co)derivations. In this paper, we investigate further properties on (X,*,&)-self-(co)derivations  and provide conditions for an (X,*,&)-self-(co)derivation to be regular. We introduce the notion of ranked *-subsystems  and investigate related properties.


## 2. Preliminaries

In a nonempty setX with a constant 0 and a binary operation *, we consider the following axioms: (a1)
((x*y)*(x*z))*(z*y)=0,(a2)
(x*(x*y))*y=0,(a3)
x*x=0,(a4)
x*y=0andy*x=0implyx=y,(b1)
x*0=x,(b2)
(x*y)*z=(x*z)*y,(b3)
((x*z)*(y*z))*(x*y)=0,(b4)
x*(x*(x*y))=x*y.IfX satisfies axioms (a1), (a2), (a3), and (a4), then we say that (X,*,0) is a BCI-algebra. Note that a BCI-algebra (X,*,0) satisfies conditions (b1), (b2), (b3), and (b4) (see [9]).In ap-semisimple BCI-algebra X, the following hold: (b5)
(x*z)*(y*z)=x*y,(b6)
0*(0*x)=x.


## 3. Derivations on Ranked Bigroupoids

Aranked bigroupoid (see [8]) is an algebraic system (X,*,•) where X is a non-empty set and “*” and “•” are binary operations defined on X. We may consider the first binary operation * as the major operation and the second binary operation • as the minor operation.Given a ranked bigroupoid(X,*,&), a map d:X→X is called an (X,*,&)-self-derivation (see [8]) if for all x,y∈X,
(3.1)d(x*y)=(d(x)*y)&(x*d(y)).
In the same setting, a map d:X→X is called an (X,*,&)-self-coderivation (see [8]) if for all x,y∈X,
(3.2)d(x*y)=(x*d(y))&(d(x)*y).
Note that if (X,*) is a commutative groupoid, then (X,*,&)-self-derivations are (X,*,&)-self-coderivations. A map d:X→X is called an abelian-(X,*,&)-self-derivation(see [8]) if it is both an (X,*,&)-self-derivation and an (X,*,&)-self-coderivation.Proposition 3.1.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y=y*(y*x) for all x,y∈X.(1)
Assume thatX satisfies axioms (b1), (b2), (b3), (a3), and (a4). If a map d:X→X is an (X,*,&)-self-derivation, then d(x)=d(x)&x for all x∈X.(2)
IfX satisfies two axioms (b1) and (a3) and a map d:X→X is an (X,*,&)-self-coderivation, then the following are equivalent:
(2.1)d(0)=0;(2.2)(∀x∈X)(d(x)=x&d(x)).Proof.
(1) Letx∈X. Using (b1) and (b2), we have
(3.3)d(x)=d(x*0)=(d(x)*0)&(x*d(0))=d(x)&(x*d(0))=(x*d(0))*((x*d(0))*d(x))=(x*d(0))*((x*d(x))*d(0)).
It follows from (b3) that
(3.4)d(x)*(d(x)&x)=((x*d(0))*((x*d(x))*d(0)))*(d(x)&x)=0.
Using (b2) and (a3), we have (d(x)&x)*d(x)=0, and so d(x)=d(x)&x for all x∈X by (a4).
(2) Letd be an (X,*,&)-self-coderivation. If  d(0)=0, then
(3.5)d(x)=d(x*0)=(x*d(0))&(d(x)*0)=x&d(x)
for all x∈X. Assume that d(x)=x&d(x) for all x∈X. Taking x=0 implies that d(0)=0&d(0)=0.Corollary 3.2.
Let(X,*,&) be a ranked bigroupoid in which (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X.(1)
If a mapd:X→X is an (X,*,&)-self-derivation, then d(x)=d(x)&x for all x∈X.(2)
If a mapd:X→X is an (X,*,&)-self-coderivation, then the following are equivalent:
(2.1)
d(0)=0;(2.2)
(∀x∈X)(d(x)=x&d(x)).Lemma 3.3.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 in which three axioms (b2),(a3), and (a4) are valid and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X.(1)
For everyx∈X with x&0=x, one has
(3.6)(∀y∈X)(y*x=0⇒y=x).(2)
If an elementa of X satisfies a&0=a, then a&x=a for all x∈X.Proof.
(1) Lety∈X be such that y*x=0. Then
(3.7)x*y=(x&0)*y=(0*y)*(0*x)=((y*x)*y)*(0*x)=(0*x)*(0*x)=0,
and so y=x by (a4).
(2) Since(a&x)*a=0, it follows from (3.6) that a&x=a for all x∈X.Corollary 3.4.
Let(X,*,&) be a ranked bigroupoid in which (X,*,0) is a BCI-algebra and the minor operation  & is defined by x&y=y*(y*x) for all x,y∈X.(1)
For everyx∈X with x&0=x, one has
(3.8)(∀y∈X)(y*x=0⇒y=x).(2)
If an elementa of X satisfies a&0=a, then a&x=a for all x∈X.Proposition 3.5.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 in which four axioms (b2),(b4),  (a3), and (a4) are valid and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. If a map d:X→X is an (X,*,&)-self-coderivation, then 0*d(x)=d(x) for all x∈X with 0*x=x.Proof.
Letx∈X be such that 0*x=x. Since (0*d(x))&0=0*d(x), it follows from Lemma 3.3(2) that d(x)=d(0*x)=(0*d(x))&(d(0)*x)=0*d(x).Corollary 3.6.
Let(X,*,&) be a ranked bigroupoid in which (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. If a map d:X→X is an (X,*,&)-self-coderivation, then 0*d(x)=d(x) for all x∈X with 0*x=x.Using Proposition3.5, we can find an (X,*,&)-self-derivation which is not an (X,*,&)-self-coderivation.Example 3.7.
Let(ℤ,-,&) be a ranked bigroupoid where ℤ is the set of all integers with the minus operation “−” and the minor operation “&” defined by x&y=y-(y-x) for all x,y∈ℤ. Let d be a self map of ℤ given by d(x)=x-1 for all x∈ℤ. Then d is a (ℤ,-,&)-self-derivation since
(3.9)d(x-y)=(x-y)-1=(x-y+1)-2=(x-y-1)&(x-y+1)=((x-1)-y)&(x-(y-1))=(d(x)-y)&(x-d(y)).
Note that 0-d(0)=0-(0-1)=1≠-1=0-1=d(0). Hence d is not a (ℤ,-,&)-self-coderivation by Proposition 3.5.Proposition 3.8.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. For an (X,*,&)-self-derivation d:X→X, if (X,*,0) satisfies axioms (b2),(b5), and (b6), then d(x)=d(0)*(0*x) for all x∈X. Moreover, if d(0)=0, then d is an identity map.Proof.
Assume that(X,*,0) satisfies axioms (b2), (b5), and (b6). Then
(3.10)d(x)=d(x&0)=(d(0)*(0*x))&(0*d(0*x))=(0*d(0*x))*((0*d(0*x))*(d(0)*(0*x)))=(0*d(0*x))*((0*(d(0)*(0*x)))*d(0*x))=0*(0*(d(0)*(0*x)))=d(0)*(0*x),
for all x∈X. Moreover, if d(0)=0 then d(x)=d(0)*(0*x)=x&0=x for all x∈X, and so d is an identity map.Corollary 3.9.
Let(X,*,&) be a ranked bigroupoid in which (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. If a map d:X→X is an (X,*,&)-self-derivation, then(1)
d(0)=d(0)&0;(2)
if(X,*,0) is p-semisimple, then d(x)=d(0)*(0*x) for all x∈X;(3)
if(X,*,0) is p-semisimple and d(0)=0, then d is an identity map.Definition 3.10.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0. A self map d of (X,*,&) is said to be regular if d(0)=0.Example 3.11.
Consider a ranked bigroupoid(X,*,&) in which X={0,a,b,c,d,e} and binary operations “*” and “&” are defined by
(3.11)x*y={0if(x,y)∈{(0,a),(b,d),(c,e)}∪{(z,z)∣z∈X},aif(x,y)∈{(a,0),(d,b),(e,c)},bif(x,y)∈{(b,0),(0,c),(0,e),(a,e),(b,a),(c,b),(c,d),(d,a),(e,d)},cif(x,y)∈{(c,0),(c,a),(e,a),(0,b),(b,c),(0,d),(a,d),(b,e),(d,e)},dif(x,y)∈{(d,0),(e,b),(a,c)},eif(x,y)∈{(a,b),(d,c),(e,0)}&0abcde0000000a0a00aabbb0bbbcccccccdbdbbddececcee.
Define a map d:X→X by
(3.12)d(x)={0ifx∈{0,a},bifx∈{b,d},cifx∈{c,e}.
Then d is an abelian (X,*,&)-self-derivation which is regular.Proposition 3.12.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y=y*(y*x) for all x,y∈X and 0*x=0 for all x∈X. Then every (X,*,&)-self-derivation is regular. Moreover, if X satisfies the axioms (b1) and (a3) then every (X,*,&)-self-coderivation is regular.Proof.
Letd be an (X,*,&)-self-derivation. Then
(3.13)d(0)=d(0*x)=(d(0)*x)&(0*d(x))=(d(0)*x)&0=0.
If d is an (X,*,&)-self-coderivation, then
(3.14)d(0)=d(0*x)=(0*d(x))&(d(0)*x)=0&(d(0)*x)=0.
Hence every (X,*,&)-self-(co)derivation is regular.Proposition 3.13.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y=y*(y*x) for all x,y∈X and two axioms (a3) and (b1) are satisfied. Let d be a self map of X and a∈X such that d(x)*a=0 (resp., a*d(x)=0) for all x∈X. If d is an (X,*,&)-self-derivation (resp., (X,*,&)-self-coderivation), then it is regular.Proof.
Assume thatd is an (X,*,&)-self-derivation. For any x∈X, we have
(3.15)0=d(x*a)*a=((d(x)*a)&(x*d(a)))*a=(0&(x*d(a)))*a=0*a,
which implies that
(3.16)d(0)=d(0*a)=(d(0)*a)&(0*d(a))=0&(0*d(a))=0.
Hence d is regular. Now, let d be an (X,*,&)-self-coderivation such that a*d(x)=0 for all x∈X. Then
(3.17)0=a*d(a*x)=a*((a*d(x))&(d(a)*x))=a*(0&(d(a)*x))=a*0,
and so
(3.18)d(0)=d(a*0)=(a*d(0))&(d(a)*0)=0&(d(a)*0)=0&d(a)=0.
Therefore d is regular.Definition 3.14.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0. Let d be a self map of (X,*,&). A subset A of X is called a ranked *-subsystem of X if it satisfies the following:(r1)
0∈A,(r2)
(∀x,y∈X)(x∈A,y*x∈A⇒y∈A).
Moreover, if a ranked*-subsystem A of X satisfies d(A)⊆A, then we say that  A is rankedd-invariant.Example 3.15.
Consider a ranked bigroupoid(X,*,&) in which X={0,a,b,c,d,e} and binary operations  “*”and “&” are defined by
(3.19)x*y={0if(x,y)∈{(0,a),(b,c),(b,d),(b,e),(c,d),(c,e)}∪{(z,z)∣z∈X},aif(x,y)∈{(a,0),(c,b),(d,b),(e,b),(d,c),(e,c),(e,d),(d,e)},cif(x,y)=(c,0),dif(x,y)=(d,0),eif(x,y)=(e,0),botherwise,
and x&y=y*(y*x) for all x,y∈X. Define a map d: X→X by
(3.20)d(x)={bifx∈{0,a}0otherwise.
Then d is an abelian (X,*,&)-self-derivation which is not regular. It is easily check that A={0,a} is a ranked *-subsystem of  X. Since d(A)={b}⊈A, d is not ranked d-invariant.Example 3.16.
In Example3.11, A={0,a} is a ranked d-invariant *-subsystem of X.Theorem 3.17.
Let(X,*,&) be a ranked bigroupoid with distinguished element 0 in which three axioms (b1),(b2), and (a3) are valid and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. For an (X,*,&)-self-coderivation d, if d is regular then every ranked *-subsystem of X is ranked d-invariant.Proof.
Assume thatd is regular and let A be a ranked *-subsystem of X. Then d(x)=x&d(x) for all x∈X by Proposition 3.1(2). Let y∈d(A). Then y=d(a) for some a∈A. Thus y*a=d(a)*a=(a&d(a))*a=0∈A, and so y∈A by (r2). Hence d(A)⊆A and A is ranked d-invariant.Corollary 3.18.
Letd be an (X,*,&)-self-coderivation where (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. If d is regular, then every ideal of X is ranked d-invariant.Example3.15 shows that Theorem 3.17 is not true if we drop the regularity of d.We consider the converse of Theorem3.17.Theorem 3.19.
Letd be an (X,*,&)-self-coderivation where (X,*,&) is a ranked bigroupoid with distinguished element 0 in which the minor operation & is defined by x&y=y*(y*x) for all x,y∈X and there does not exist a nonzero element x of X such that x*0=0. If every ranked *-subsystem of X is ranked d-invariant, then d is regular.Proof.
Assume that every ranked*-subsystem of X is ranked d-invariant. Note that A={0} is a ranked *-subsystem of X. Thus d(A)=d({0})⊆{0}, and therefore d(0)=0,thatis,d is regular.Corollary 3.20.
Letd be an (X,*,&)-self-coderivation where (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. Then d is regular if and only if every ranked *-subsystem of X is ranked d-invariant.Proposition 3.21.
Let(X,*,&) be a ranked bigroupoid where (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. For any α∈X, let dα be a self map of X defined by dα(x)=x*α for all x∈X. If X satisfies the following conditions:(1)
((x*y)*z)*(x*(y*z))=0 for all x,y,z∈X,(2)
(∀x,y∈X)(x*y=0⇒x=y), then dα is an abelian (X,*,&)-self-derivation.Proof.
IfX satisfies two given conditions, then the following identity is valid (see [9]):
(3.21)(∀x,y,z∈X)((x*y)*z=x*(y*z)).
It follows from (b1), (a3), and (b2) that
(3.22)dα(x*y)=(x*y)*α=(x*(y*α))*0=(x*(y*α))*((x*(y*α))*(x*(y*α)))=(x*(y*α))*((x*(y*α))*((x*α)*y))=(dα(x)*y)&(x*dα(y)).
Hence dα is an (X,*,&)-self-derivation. Similarly, we can verify that dα is an (X,*,&)-self-coderivation.Corollary 3.22.
Let(X,*,&) be a ranked bigroupoid where (X,*,0) is a BCI-algebra and the minor operation & is defined by x&y=y*(y*x) for all x,y∈X. For any α∈X, let dα be a self map of X defined by dα(x)=x*α for all x∈X. If X satisfies (b1) and the following conditions: (1)
((x*y)*z)*(x*(y*z))=0 for all x,y,z∈X,(2)
(x*y)*(x*z)=z*y for all x,y,z∈X, then dα is an abelian (X,*,&)-self-derivation.Proof.
IfX satisfies both (b1) and the second condition, then X is a p-semisimple BCI-algebra (see [9]). Hence the second condition of Proposition 3.21 is valid. Therefore dα is an abelian (X,*,&)-self-derivation.


## 4. Conclusion

Alshehri et al. [8] introduced the notion of ranked bigroupoids and discussed (X,*,&)-self-(co)derivations.A nonempty setX together with maps *:X×X→X and &:X×X→X is called a ranked bigroupoid. For a ranked bigroupoid (X,*,&), a map d:X→X is called: (1)
an(X,*,&)-self-derivation if
(4.1)d(x*y)=(d(x)*y)&(x*d(y))
for all x,y∈X;(2)
an(X,*,&)-self-coderivation if
(4.2)d(x*y)=(x*d(y))&(d(x)*y)
for all x,y∈X.In this paper, we have investigated further properties on(X,*,&)-self-(co)derivations and have provided conditions for an (X,*,&)-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,Γ-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.



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