# f

Authors: Deena Al-Kadi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

---

## Abstract

We introduce the notion off
q-derivation as a new derivation of G-algebra. For an endomorphism map f of any G-algebra X
, we show that at least one f
q-derivation of X exists. Moreover, for such a map, we show that a self-map d
q
f of X is f
q-derivation of X if X is an associative medial G-algebra. For a medial G-algebra X, d
q
f is f
q-derivation of X if d
q
f is an outside f
q-derivation of X. Finally, we show that if f is the identity endomorphism of X then the composition of two f
q-derivations of X is a f
q-derivation. Moreover, we give a condition to get a commutative composition.


---

## Body

## 1. Introduction

Derivation is an important area of research in the theory of algebraic structure in mathematics. The theory of derivations of algebraic structures came from the development of Galois theory and the theory of invariants. Many researches have been done on derivations on different algebras (see [1–4]).Several authors [5–9] have studied derivations in B
C
I-algebra after the work done in 2004 by Jun and Xin where the notion of derivation in ring and near-ring theory was applied to B
C
I-algebra [4]. As in [5], for a self-map d, f
o
r
a
n
y
a
l
g
e
b
r
a
X, d is a left-right derivation (briefly l
,
r-derivation) of X if it satisfies the identity d
x
∗
y
=
d
x
∗
y
∧
x
∗
d
y for all x
,
y
∈
X
. If d satisfies the identity d
(
x
∗
y
)
=
(
x
∗
d
(
y
)
)
∧
(
d
(
x
)
∗
y
) for all x
,
y
∈
X, then d is a right-left derivation (briefly r
,
l-derivation) of X
. If d is both l
,
r- and r
,
l-derivation, then d is a derivation of X
.Recently, in 2013, a new derivation namedf
q-derivation of B
C
I-algebras was introduced. That is, in general, for any self-map d
q
f   of an algebra X, f
q-derivation of X is defined by d
q
f
(
x
)
=
f
x
∗
q for all x and  q
∈
X. The map d
q
f is called an outside f
q-derivation of X if it satisfies d
q
f
x
∗
y
=
f
x
∗
d
q
f
y
∧
d
q
f
x
∗
f
y
,
∀
x
,
y
∈
X
. If the map d
q
f satisfies the identity d
q
f
x
∗
y
=
d
q
f
x
∗
f
y
∧
f
x
∗
d
q
f
y
,
∀
x
,
y
∈
X, then the map d
q
f is called an inside f
q-derivation of X. If d
q
f is both outside and inside f
q-derivation of X, then it is a f
q-derivation of X ([10]).The notion ofG-algebra was introduced in [11]. The aim of the paper is to complete the studies on G-algebra; in particular, we aim to apply the notion of f
q-derivation on G-algebra and obtain some related properties. We start with definitions and propositions on G-algebra taken from [11]. Then, we redefine the notion of f
q-derivation in G-algebra and prove that every self-map d
q
f of an associative, medial G-algebra is f
q-derivation, where f is an endomorphism of X. We also show that every self-map d
q
f of an associative, medial G-algebra is f
q-derivation. Then, we show that if f is the identity endomorphism of X, then, for a medial G-algebra, d
q
f is a f
q-derivation of X if d
q
f is an outside f
q-derivation of X. Further, we show that if f is the identity endomorphism of X and d
q
f, d
q
′
f are both outside (resp., inside) f
q-derivations of X, then the composition is an outside (resp., inside) f
q-derivation of X and consequently f
q-derivation. We conclude the section with a condition given on two f
q-derivations of X to get a commutative composition.Definition 1.
AG-algebra is a nonempty set X with a constant 0 and a binary operation ∗ satisfying the axioms:(1)
x
∗
x
=
0,
(2)
x
∗
(
x
∗
y
)
=
y, for all x, y in X.Proposition 2.
If(
X
,
∗
,
0
) is a G-algebra, then the following conditions hold:(1)
x
∗
0
=
x,
(2)
0
∗
(
0
∗
x
)
=
x, for any x
∈
X
.Proposition 3.
Let(
X
,
∗
,
0
) be a G-algebra. Then, the following conditions hold for any x
,
y
∈
X: (1)
(
x
∗
x
∗
y
)
∗
y
=
0,
(2)
x
∗
y
=
0
⇒
x
=
y,
(3)
0
∗
x
=
0
∗
y
⇒
x
=
y
.Definition 4.
AG-algebra X satisfying x
∗
y
∗
z
∗
u
=
x
∗
z
∗
y
∗
u, for any x
,
y
,
z and u
∈
X, is called a medial G-algebra.Lemma 5.
IfX is a medial G-algebra, then, for any x
,
y
,
z
∈
X, the following axiom holds:(1)
x
∗
y
∗
z
=
x
∗
z
∗
y
.Theorem 6.
AG-algebra X is medial if and only if it satisfies the following conditions:(1)
y
∗
x
=
0
∗
x
∗
y for all x
,
y
∈
X,
(2)
x
∗
y
∗
z
=
z
∗
y
∗
x for all x
,
y
,
z
∈
X
.


## 2. Results

In this section we will introduce a new derivation ofG-algebra motivated by [10, Definition  3.1]. We start by defining an endomorphism of G-algebra X.Definition 7.
LetX be a G-algebra and let f be a self-map of X. One says that f is an endomorphism if (2)
f
x
∗
y
=
f
x
∗
f
y
,
∀
x
,
y
∈
X
.Throughout the paper, d
q
f is a self-map of G-algebra X defined by d
q
f
(
x
)
=
f
x
∗
q for all x
∈
X, q
∈
X and f is an endomorphism self-map of X unless otherwise mentioned.For elementsx and y of a G-algebra X, denote x
∧
y by y
∗
y
∗
x
. By considering that x
∧
y
=
x in G-algebra, we redefine the notion of f
q-derivation in [10] to get the following definition.Definition 8.
A mapd
q
f is called an outside f
q-derivation of X if(3)
d
q
f
x
∗
y
=
f
x
∗
d
q
f
y
,
∀
x
,
y
∈
X
.If the map d
q
f satisfies the following identity:(4)
d
q
f
x
∗
y
=
d
q
f
x
∗
f
y
,
∀
x
,
y
∈
X
,then the map is called an inside f
q-derivation of X. If d
q
f is both an outside and inside f
q-derivation of X, then d
q
f is a f
q-derivation of X.Remark 9.
Ifd
q
f is f
q-derivation, then(5)
d
q
f
x
∗
y
=
f
x
∗
d
q
f
y
=
d
q
f
x
∗
f
y
.Example 10.
Consider theG-algebra given by Cayley table (Table 1).
Define an endomorphism:(6)
f
:
X
⟶
X
,
such that
x
⟼
0
if
x
=
0
,
2
if
x
=
1
,
1
if
x
=
2
.If q
=
0, then Table 2 shows that d
0
f is an outside and an inside f
q-derivation of X
. Hence, d
0
f is a f
q-derivation of X
.
If we takeq
=
2, then d
2
f is not an outside f
q-derivation of X or an inside f
q-derivation of X since d
2
f
1
∗
2
=
2
w
h
i
l
e
f
1
∗
d
2
f
2
=
0 and d
2
f
1
∗
f
2
=
1
.Table 1
∗
0
1
2
0
0
1
2
1
1
0
2
2
2
1
0Table 2
x
0
0
0
1
1
1
2
2
2
y
0
1
2
0
1
2
0
1
2
x
∗
y
0
1
2
1
0
2
2
1
0
f
(
x
∗
y
)
0
2
1
2
0
1
1
2
0
d
0
f
(
x
∗
y
)
0
2
1
2
0
1
1
2
0
f
(
x
)
0
0
0
2
2
2
1
1
1
d
0
f
(
x
)
0
0
0
2
2
2
1
1
1
f
(
y
)
0
2
1
0
2
1
0
2
1
d
0
f
(
y
)
0
2
1
0
2
1
0
2
1
f
(
x
)
∗
d
0
f
(
y
)
0
2
1
2
0
1
1
2
0
d
0
f
(
x
)
∗
f
(
y
)
0
2
1
2
0
1
1
2
0Example 11.
LetX
=
0
,
a
,
b
,
c. Consider the G-algebra given by Cayley table (Table 3).
Define an endomorphism:(7)
f
:
X
⟶
X
,
such that
x
⟼
0
if
x
=
0
,
b
if
x
=
a
,
a
if
x
=
b
,
c
if
x
=
c
.It can be shown by direct calculation that d
q
f is f
q-derivation of X for all q
∈
X
.Table 3
∗
0
a
b
c
0
0
a
b
c
a
a
0
c
b
b
b
c
0
a
c
c
b
a
0Proposition 12.
For anyG-algebra X, there exists at least one f
q-derivation of X, that is, the map d
0
f
.Proof.
Letq
=
0;  t
h
e
n
d
0
f
x
∗
y
=
f
x
∗
y
∗
0
=
f
x
∗
f
y
∗
0
=
f
(
x
)
∗
f
(
y
) and f
x
∗
d
0
f
y
=
f
x
∗
f
y
∗
0
=
f
(
x
)
∗
f
(
y
)
. We also have d
0
f
x
∗
f
y
=
f
x
∗
0
∗
f
(
y
)
=
f
(
x
)
∗
f
(
y
)
. Hence, d
0
f is f
q-derivation of X
.Proposition 13.
IfX is an associative G-algebra, then d
q
f is an outside f
q-derivation of X, for all q
∈
X
.Proof.
We haved
q
f
x
∗
y
=
f
x
∗
y
∗
q
=
f
x
∗
f
y
∗
q and f
x
∗
d
q
f
y
=
f
x
∗
f
y
∗
q
=
f
x
∗
f
y
∗
q, as X is associative. Hence, d
q
f is an outside f
q-derivation of X.Proposition 14.
IfX is a medial G-algebra, then d
q
f is an inside f
q-derivation of X, for all q
∈
X
.Proof.
Sinced
q
f
x
∗
y
=
f
x
∗
y
∗
q
=
f
x
∗
f
y
∗
q and d
q
f
x
∗
f
y
=
f
x
∗
q
∗
f
(
y
)
=
f
x
∗
f
y
∗
q, as X is medial, therefore, d
q
f is an inside f
q-derivation of X.The next theorem follows from Propositions13 and 14.Theorem 15.
LetX be an associative medial G-algebra; then d
q
f is a f
q-derivation of X, for all q
∈
X
.Next we provide an alternative proof of Theorem15.Theorem 16.
LetX be an associative medial G-algebra. Then, d
q
f is both an outside f
q-derivation of X and an inside f
q-derivation of X for any q
∈
X.Proof.
Letx
,
y
,
q
∈
X. Then, on one hand, we have(8)
d
q
f
x
∗
y
=
f
x
∗
y
∗
q
=
f
x
∗
f
y
∗
q
∗
0
=
f
x
∗
q
∗
f
y
∗
0
=
f
x
∗
q
∗
f
y
∗
f
x
∗
q
∗
f
y
∗
f
x
∗
q
∗
f
y
=
d
q
f
x
∗
f
y
∗
d
q
f
x
∗
f
y
∗
f
x
∗
f
y
∗
q
=
d
q
f
x
∗
f
y
∗
d
q
f
x
∗
f
y
∗
f
x
∗
f
y
∗
q
=
d
q
f
x
∗
f
y
∗
d
q
f
x
∗
f
y
∗
f
x
∗
d
q
f
y
=
f
x
∗
d
q
f
y
,
a
s
y
∗
y
∗
x
=
x
. By Definition 8
,
d
q
f is an outside f
q-derivation of X.
On the other hand,(9)
d
q
f
x
∗
y
=
f
x
∗
y
∗
q
=
f
x
∗
y
∗
q
∗
0
=
f
x
∗
y
∗
q
∗
f
x
∗
y
∗
q
∗
f
x
∗
y
∗
q
=
f
x
∗
f
y
∗
q
∗
f
x
∗
f
y
∗
q
∗
f
x
∗
f
y
∗
q
=
f
x
∗
f
y
∗
q
∗
f
x
∗
f
y
∗
q
∗
f
x
∗
q
∗
f
y
=
f
x
∗
d
q
f
y
∗
f
x
∗
d
q
f
y
∗
d
q
f
x
∗
f
y
=
d
q
f
x
∗
f
y
.Therefore, d
q
f is an inside f
q-derivation of X
.Using Proposition3
(
2
), we get the following.Proposition 17.
Ifd
q
f is an outside (resp., inside) f
q-derivation of X, then d
q
f
0
=
f
x
∗
d
q
f
x
,
∀
x
∈
X  (resp., d
q
f
0
=
d
q
f
x
∗
f
x
,
∀
x
∈
X
).Proof.
It is obvious.Theorem 18.
LetX be a medial G-algebra. If d
q
f is an outside f
q-derivation of X, then d
q
f is a f
q-derivation of X.Proof.
From Proposition14, we know that d
q
f is an inside f
q-derivation of X. Thus, d
q
f is a f
q-derivation of X.Definition 19.
A mapd
q
f is said to be regular if d
q
f
0
=
0
.Proposition 20.
Letd
q
f be a f
q-derivation of X
. If either f
x
∗
d
q
f
y
=
0 or d
q
f
x
∗
f
y
=
0, then d
q
f is a regular derivation.Proof.
Sinced
q
f is a f
q-derivation, we have d
q
f
x
∗
y
=
f
x
∗
d
q
f
y
=
d
q
f
x
∗
f
y
. Consider that f
x
∗
d
q
f
y
=
0; then d
q
f
0
=
d
q
f
x
∗
x
=
f
x
∗
d
q
f
x
=
0
. Similarly, if d
q
f
x
∗
f
y
=
0, we have d
q
f
0
=
d
q
f
x
∗
x
=
d
q
f
x
∗
f
x
=
0
. This proves that d
q
f is a regular derivation.Proposition 21.
Letd
q
f be a regular f
q-derivation of X; then d
q
f
x
=
f
x
,
∀
x
∈
X
.Proof.
Sinced
q
f is a regular f
q-derivation of X, then d
q
f
0
=
0
. So d
q
f
0
=
d
q
f
x
∗
x
=
d
q
f
x
∗
f
x
=
0
. Therefore, d
q
f
x
=
f
x from Proposition 3
(
2
).Definition 22.
LetX be a G-algebra and let d
q
f
,
d
q
′
f be two self-maps of X. Define d
q
f
∘
d
q
′
f
:
X
→
X by(10)
d
q
f
∘
d
q
′
f
x
=
d
q
f
d
q
′
f
x
∀
x
∈
X
.Proposition 23.
LetX be a G-algebra and let f be the identity endomorphism of X. If d
q
f
,
d
q
′
f are outside f
q-derivations of X, then d
q
f
∘
d
q
′
f is also an outside f
q-derivation.Proof.
Consider the elementx
∗
y. Then d
q
f
∘
d
q
′
f
x
∗
y
=
d
q
f
d
q
′
f
x
∗
y. As d
q
′
f
a
n
d
d
q
f are outside f
q-derivations, we have d
q
f
d
q
′
f
x
∗
y
=
d
q
f
f
x
∗
d
q
′
f
(
y
)
=
f
x
∗
d
q
f
∘
d
q
′
f
y
. Thus, d
q
f
∘
d
q
′
f is an outside f
q-derivation.Similarly, we can prove the following proposition.Proposition 24.
For aG-algebra X, let f be the identity endomorphism of X. If d
q
f
,
d
q
′
f are inside f
q-derivations of X, then d
q
f
∘
d
q
′
f is also an inside f
q-derivation.Combining Proposition23 and Proposition 24 we have the following theorem.Theorem 25.
LetX be a G-algebra and let f be the identity endomorphism of X. If d
q
f
,
d
q
′
f are both outside (resp., inside) f
q-derivations of X, then the composition is an outside (resp., inside) f
q-derivation of X.Proposition 26.
LetX be a G-algebra and let d
q
f
,
d
q
′
f be f
q-derivations of X such that d
q
f
∘
f
=
f
∘
d
q
f and d
q
′
f
∘
f
=
f
∘
d
q
′
f; then d
q
f
∘
d
q
′
f
=
d
q
′
f
∘
d
q
f.Proof.
Considerd
q
′
f as an outside f
q-derivation of X and d
q
f as an inside f
q-derivation of X; then for all x
,
y
∈
X we have(11)
d
q
f
∘
d
q
′
f
x
∗
y
=
d
q
f
d
q
′
f
x
∗
y
=
d
q
f
f
x
∗
d
q
′
f
y
=
d
q
f
f
x
∗
f
d
q
′
f
y
=
d
q
f
∘
f
x
∗
f
∘
d
q
′
f
y
.On the other hand,(12)
d
q
′
f
∘
d
q
f
x
∗
y
=
d
q
′
f
d
q
f
x
∗
y
=
d
q
′
f
d
q
f
x
∗
f
y
=
f
d
q
f
x
∗
d
q
′
f
f
y
=
f
∘
d
q
f
x
∗
d
q
′
f
∘
f
y
=
d
q
f
∘
f
x
∗
f
∘
d
q
′
f
y
.From (11) and (12), we can see that d
q
f
∘
d
q
′
f
x
∗
y
=
d
q
′
f
∘
d
q
f
x
∗
y
. By putting y
=
0, we get (13)
d
q
f
∘
d
q
′
f
x
=
d
q
′
f
∘
d
q
f
x
.Hence, d
q
f
∘
d
q
′
f
=
d
q
′
f
∘
d
q
f
.


## 3. Conclusion

In this paper, the notion off
q-derivation of G-algebra is introduced and some related properties are investigated. The main results are Theorems 15 and 18 where we show that a self-map d
q
f of a G-algebra is f
q-derivation if G-algebra satisfies some properties. In Theorem 25, we show that in G-algebra the composition of two f
q-derivations of X is f
q-derivation if f is identity endomorphism of X
. Moreover, we give in Proposition 26 a condition on two f
q-derivations of X to get a commutative composition.



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*Source: 9276096-2016-04-28.xml*