The notion of generalized (regular) (α, β) derivations of a BCIalgebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a generalized (α, β) derivation to be regular is provided. The concepts of a generalized Finvariant (α, β) derivation and αideal are introduced, and their relations are discussed. Moreover, some results on regular generalized (α, β) derivations are proved..
AMS Mathematics Subject Classification : 03G25, 06F35.
1. Introduction
Throughout our discussion X will denote a BCIalgebra unless otherwise mentioned. In the year 2004, Jun and Xin
[1]
applied the notion of derivation in ring and nearring theory to
BCI
algebras, and as a result they introduced a new concept, called a (regular) derivation, in
BCI
algebras. Using this concept as defined they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of a
p
semisimple
BCI
algebra. For a self map
d
of a
BCI
algebra, they defined a
d
invariant ideal, and gave conditions for an ideal to be
d
invariant. According to Jun and Xin, a self map
d
:
X
→
X
is called a leftright derivation (briefly (
l
,
r
)derivation)of
X
if
d
(
x
∗
y
) =
d
(
x
) ∗
y
∧
x
∗
d
(
y
) holds for all
x
,
y
∈
X
. Similarly, a self map
d
:
X
→
X
is called a rightleft derivation (briefly (
r
,
l
)derivation) of
X
if
d
(
x
∗
y
) =
x
∗
d
(
y
)∧
d
(
x
)∗
y
holds for all
x
,
y
∈
X
. Moreover, if
d
is both (
l
,
r
)−and (
r
,
l
)−derivation, it is a derivation on
X
. After the work of Jun and Xin
[1]
, many research articles have appeared on the derivations of BCIalgebras and a greater interest have been devoted to the study of derivations in BCI algebras on various aspects (see
[2
,
3
,
4
,
5
,
6
,
7]
).
Recently in
[5]
, Muhiuddin and Alroqi introduced the notion of (
α
,
β
)derivations of a
BCI
algebra, and investigated some related properties. Using the idea of
regular
(
α
,
β
)
derivations
, they gave characterizations of a
p
semisimple
BCI
algebra. In the present paper, we consider a more general version of the paper
[5]
. We first introduce the notion of
generalized (regular)
(
α
,
β
)
derivations
of a
BCI
algebra, and investigate related properties. We provide a condition for a
generalized
(
α
,
β
)
derivation
to be regular. We also introduce the concepts of a
generalized Finvariant
(
α
,
β
)
derivation
and
α
ideal, and then we investigate their relations. Furthermore, we obtain some results on regular
generalized
(
α
,
β
)
derivations
.
2. Preliminaries
We begin with the following definitions and properties that will be needed in this paper.
A nonempty set
X
with a constant 0 and a binary operation ∗ is called a
BCIalgebra
if for all
x
,
y
,
z
∈
X
the following conditions hold:

(I) ((x∗y) ∗ (x∗z)) ∗ (z∗y) = 0,

(II) (x∗ (x∗y)) ∗y= 0,

(III)x∗x= 0,

(IV)x∗y= 0 andy∗x= 0 implyx=y.
Define a binary relation ≤ on
X
by letting
x
∗
y
= 0 if and only if
x
≤
y
. Then (
X
,≤) is a partially ordered set. A BCIalgebra
X
satisfying 0 ≤
x
for all
x
∈
X
, is called BCKalgebra.
A
BCI
algebra
X
has the following properties: for all
x
;
y
;
z
∈
X

(a1)x∗ 0 =x.

(a2) (x∗y) ∗z= (x∗z) ∗y,

(a3)x≤yimpliesx∗z≤y∗zandz∗y≤z∗x.

(a4) (x∗z) ∗ (y∗z) ≤x∗y,

(a5)x∗ (x∗ (x∗y)) =x∗y,

(a6) 0 ∗ (x∗y) = (0 ∗x) ∗ (0 ∗y),

(a7)x∗ 0 = 0 impliesx= 0.
For a
BCI
algebra
X
, denote by
X
_{+}
(resp.
G
(
X
)) the
BCK
part (resp. the
BCI
G part) of
X
, i.e.,
X
_{+}
is the set of all
x
∈
X
such that 0 ≤
x
(resp.
G
(
X
) := {
x
∈
X
 0 ∗
x
=
x
}). Note that
G
(
X
) ∩
X
_{+}
= {0} (see
[8]
). If
X
_{+}
= {0}, then
X
is called a
psemisimple BCIalgebra
. In a
p
semisimple
BCI
algebra
X
, the following hold:

(a8) (x∗z) ∗ (y∗z) =x∗y.

(a9) 0 ∗ (0 ∗x) =xfor allx∈X.

(a10)x∗ (0 ∗y) =y∗ (0 ∗x).

(a11)x∗y= 0 impliesx=y.

(a12)x∗a=x∗bimpliesa=b.

(a13)a∗x=b∗ximpliesa=b.

(a14)a∗ (a∗x) =x.

(a15) (x∗y) ∗ (w∗z) = (x∗w) ∗ (y∗z).
Let
X
be a
p
semisimple
BCI
algebra. We define addition "+" as
x
+
y
=
x
∗ (0 ∗
y
) for all
x
,
y
∈
X
. Then (
X
, +) is an abelian group with identity 0 and
x
−
y
=
x
∗
y
. Conversely let (
X
, +) be an abelian group with identity 0 and let
x
∗
y
=
x
−
y
. Then
X
is a
p
semisimple
BCI
algebra and
x
+
y
=
x
∗ (0 ∗
y
) for all
x
,
y
∈
X
(see
[9]
).
For a
BCI
algebra
X
we denote
x
∧
y
=
y
∗(
y
∗
x
), in particular 0∗(0∗
x
) =
a_{x}
, and
L_{p}
(
X
) := {
a
∈
X

x
∗
a
= 0 ⇒
x
=
a
, ∀
x
∈
X
}. We call the elements of
L_{p}
(
X
) the
patoms
of
X
. For any
a
∈
X
, let
V
(
a
) := {
x
∈
X

a
∗
x
= 0}, which is called the
branch
of
X
with respect to
a
. It follows that
x
∗
y
∈
V
(
a
∗
b
) whenever
x
∈
V
(
a
) and
y
∈
V
(
b
) for all
x
,
y
∈
X
and all
a
,
b
∈
L_{p}
(
X
). Note that
L_{p}
(
X
) = {
x
∈
X

a_{x}
=
x
}; which is the
p
semisimple part of
X
, and
X
is a
p
semisimple
BCI
algebra if and only if
L_{p}p
(
X
) =
X
(see
[10]
,[Proposition 3.2]). Note also that
a_{x}
∈
L_{p}
(
X
), i.e., 0 ∗ (0 ∗
a_{x}
) =
a_{x}
, which implies that
a_{x}
∗
y
∈
L_{p}
(
X
) for all
y
∈
X
. It is clear that
G
(
X
) ⊂
L_{p}p
(
X
), and
x
∗ (
x
∗
a
) =
a
and
a
∗
x
∈
L_{p}
(
X
) for all
a
∈
L_{p}
(
X
) and all
x
∈
X
. For more details, refer to
[11
,
12
,
1
,
10
,
8
,
9]
.
Definition 2.1
(
[6]
). A
BCI
algebra
X
is said to be
torsion free
if it satiafies:
Definition 2.2
(
[5]
). Let
α
and
β
are two endomorphisms of a
BCI
algebra
X
. Then a self map
d
_{(α, β)}
:
X
→
X
is called a (
α
,
β
)derivation of
X
if it satisfies:
3. Main results
In what follows,
α
and
β
are endomorphisms of a
BCI
algebra
X
unless otherwise specified.
Definition 3.1.
Let
X
be a
BCK
/
BCI
algebra. Then a self map
F
on
X
is called a
generalized
(
α
,
β
)
derivation
if there exists an (
α
,
β
)
derivation d
_{(α, β)}
of
X
such that
Clearly, the notion of
generalized
(
α
,
β
)
derivation
covers the concept of (
α
,
β
)
derivation
when
F
=
d
_{(α, β)}
and the concept of
generalized derivation
when
F
=
d
_{(α, β)}
=
D
, and
α
=
β
=
I_{X}
where
I_{X}
is the identity map on
X
.
Example 3.2.
Consider a
BCI
algebra
X
= {0,
a
,
b
} with the following Cayley table:
(1) Define a map
and define two endomorphisms
and
Then
d
_{(α, β)}
is a (
α
,
β
)derivation of
X
[5]
.
Again, define a self map
It is routine to verify that
F
is a
generalized
(
α
,
β
)
derivation
of
X
.
(2) Define a map
and define two endomorphisms
and
Then
d
_{(α, β)}
is a (
α
,
β
)derivation of
X
[5]
.
Again, define a self map
F
:
X
→
X
by
F
(
x
) =
b
for all
x
∈
X
. It is routine to verify that
F
is a
generalized
(
α
,
β
)
derivation
of
X
.
Lemma 3.3
(
[12]
).
Let X be a BCIalgebra. For any x, y ∈ X, if x ≤ y, then x and y are contained in the same branch of X
.
Lemma 3.4
(
[12]
).
Let X be a BCIalgebra. For any x, y ∈ X, if x and y are contained in the same branch of X, then x ∗ y, y ∗ x ∈ X_{+}
.
Proposition 3.5.
Let X be a commutative BCIalgebra. Then every generalized (α, β)derivation F of X satisfies the following assertion
:
that is, every generalized (α, β)derivation of X is isotone
.
Proof
. Let
x
,
y
∈
X
be such that
x
≤
y
. Since
X
is commutative, we have
x
=
x
∧
y
. Hence
Since every endomorphism of
X
is isotone, we have
α
(
x
) ≤
α
(
y
): It follows from Lemma 3.3 that 0 =
α
(
x
) ∗
α
(
y
) ∈
X
_{+}
and
α
(
y
) ∗
α
(
x
) ∈
X
_{+}
so that there exists
a
(≠ 0) ∈
X
_{+}
such that
α
(
y
∗
x
) =
α
(
y
) ∗
α
(
x
) =
a
. Hence (3.3) implies that
F
(
x
) ≤
F
(
y
) ∗
a
: Using (a3), (a2) and (III), we have
and so
F
(
x
) ∗
F
(
y
) = 0, that is,
F
(
x
) ≤
F
(
y
) by (a7).
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.6
(
[5]
).
Let X be a commutative BCIalgebra
.
Then every
(
α
,
β
)
derivation d
_{(α, β)}
of X satisfies the following assertion
:
that is
,
every
(
α
,
β
)
derivation of X is isotone
.
Proposition 3.7.
Every generalized
(
α
,
β
)
derivation F of a BCIalgebra X satisfies the following assertion
:
Proof
. Let
F
be a generalized (
α
,
β
)derivation of
X
. Using (a2) and (a4), wehave
Obviously
F
(
x
) ∧
d
_{(α, β)}
(0) ≤
F
(
x
) by (II). Therefore the equality (3.5) is valid.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.8
(
[5]
).
Every
(
α
,
β
)
derivation d
_{(α, β)}
of a BCIalgebra X satisfies the following assertion:
Theorem 3.9.
Let F be a generalized
(
α
,
β
)
derivation on a BCIalgebra X
.
Then

(1) (∀a∈Lp(X),x∈X) (F(a∗x) =F(a) ∗α(x)).

(2) (∀a∈Lp(X),x∈X) (F(a+x) =F(a) +α(x)).

(3) (∀a,b∈Lp(X)) (F(a+b) =F(a) +α(b)).
Proof
. (1) For any
a
∈
Lp
(
X
), we have
a
∗
x
∈
Lp
(
X
) for all
x
∈
X
. Thus
F
(
a
∗
x
) =
F
(
a
) ∗
α
(
x
) ∧
d
_{(α, β)}
(
x
) ∗
β
(
a
) =
F
(
a
) ∗
α
(
x
).
(2) For any
a
∈
Lp
(
X
) and
x
∈
X
, it follows from (1) that
(3) The proof follows directly from (2).
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.10
(
[5]
).
Let d
_{(α, β)}
be an (
α
,
β
)
derivation on a BCIalgebra X
.
Then

(1) (∀a∈Lp(X),x∈X)(d(α,β)(a∗x) =d(α,β)(a) ∗α(x)).

(1) (∀a∈Lp(X),x∈X)(d(α,β)(a+x) =d(α,β)(a) +α(x)).

(3) (∀a,b∈Lp(X))(d(α,β)(a+b) =d(α,β)(a) +α(b)).
Definition 3.11.
Let
X
be a BCIalgebra and
F
,
F′
be two self maps of
X
, we define
F
◦
F′
:
X
→
X
by (
F
◦
F′
)(
x
) =
F
(
F′
(
x
)) for all
x
∈
X
.
Theorem 3.12.
Let X be a psemisimple BCIalgebra
.
Let F and F′ be two generalized
(
α
,
β
)
derivations associated with d
_{(α, β)}
and
(
α
,
β
)
derivations respectively on X such that α
^{2}
=
α
.
Then F
◦
F′ is an
(
α
,
β
)
derivation on X
.
Proof
. For any
x
,
y
∈
X
, it follows from (a14) that
This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.13
(
[5]
).
Let X be a psemisimple BCIalgebra
.
If d
_{(α, β)}
and
are two
(
α
,
β
)
derivations on X such that α
^{2}
=
α
,
then
is an
(
α
,
β
)
derivation on X
.
Theorem 3.14.
Let α
,
β be two endomorphisms and F be a self map on a psemisimple BCIalgebra X such that F
(
x
) =
α
(
x
)
for all x
∈
X
.
Then F is a generalized
(
α
,
β
)
derivation on X
.
Proof
. Let us take
F
(
x
) =
α
(
x
) for all
x
∈
X
. Since
x
,
y
∈
X
⇒
x
∗
y
∈
X
, by using (a14) we have
This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.15
(
[5]
).
Let α
,
β be two endomorphisms and d
_{(α, β)}
be a self map on a psemisimple BCIalgebra X such that d
_{(α, β)}
(
x
) =
α
(
x
)
for all x
∈
X
.
Then d
_{(α, β)}
is an
(
α
,
β
)
derivation on X
.
Definition 3.16.
A generalized (
α
,
β
)derivation
F
of a
BCI
algebra
X
is said to be
regular
if
F
(0) = 0:
Example 3.17.
(1) The generalized (
α
,
β
)derivation
F
of
X
in Example 3.2(1) is regular.
(2) The generalized (
α
,
β
)derivation
F
of
X
in Example 3.2(2) is not regular.
We provide a condition for a generalized (
α
,
β
)derivation to be regular.
Theorem 3.18.
Let F be a generalized
(
α
,
β
)
derivation of a BCIalgebra X
.
If there exists a
∈
X such that F
(
x
) ∗
α
(
a
) = 0
for all x
∈
X
,
then F is regular
.
Proof
. Assume that there exists
a
∈
X
such that
F
(
x
) ∗
α
(
a
) = 0 for all
x
∈
X
. Then
and so
F
(0) =
F
(0 ∗
a
) = (
F
(0) ∗
α
(
a
)) ∧(
d
_{(α, β)}
(
a
) ∗
β
(0)) = 0. Hence
F
is regular.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.19
(
[5]
).
Let d
_{(α, β)}
be an
(
α
,
β
)
derivation of a BCIalgebra X
.
If there exists a
∈
X such that d
_{(α, β)}
(
x
) ∗
α
(
a
) = 0
for all x
∈
X
,
then d
_{(α, β)}
is regular
.
Definition 3.20.
For a generalized (
α
,
β
)derivation
F
of a
BCI
algebra
X
, we say that an ideal
A
of
X
is an
α

ideal
(resp.
β

ideal
) if
α
(
A
) ⊆
A
(resp.
β
(
A
) ⊆
A
).
Defiition 3.21.
For a generalized (
α
,
β
)derivation
F
of a
BCI
algebra
X
, we say that an ideal
A
of
X
is
Finvariant
if
F
(
A
) ⊆
A
.
Example 3.22.
(1) Let
F
be a generalized (
α
,
β
)derivation of
X
which is described in Example 3.2(1). We know that
A
:= {0,
a
} is both an
α
ideal and a
β
ideal of
X
. Furthermore,
A
:= {0,
a
} is also
F
invariant.
(2) Let
F
be a generalized (
α
,
β
)derivation of
X
which is described in Example 3.2(2). We know that
A
:= {0,
a
} is a
β
ideal of
X
. But
A
:= {0,
a
} is an ideal of
X
which is neither
α
ideal nor
F
invariant.
Next, we prove some results on regular generalized (
α
,
β
)derivations in a
BCI
algebra. In our further discussion, we shall assume that for every regular generalized (
α
,
β
)derivation
F
:
X
→
X
there exists a regular (
α
,
β
)derivation
d
_{(α, β)}
:
X
→
X
i.e.
d
_{(α, β)}
(0) = 0.
Theorem 3.23.
Let F be a regular generalized
(
α
,
β
)
derivation of a BCIalgebra X
.
Then

(1) (∀a∈X) (a∈Lp(X) ⇒F(a) ∈Lp(X)).

(2) (∀a∈X) (a∈Lp(X) ⇒α(a),β(a) ∈Lp(X)).

(3) (∀a∈Lp(X)) (F(a) =F(0) +α(a)).

(4) (∀a,b∈Lp(X)) (F(a+b) =F(a) +F(b) −F(0)).
Proof
. (1) Let
F
be a regular generalized (
α
,
β
)derivation. Then the proof follows directly from Proposition 3.7.
(2) Let
a
∈
L_{p}
(
X
): Then
a
= 0 ∗ (0 ∗
a
), and so
α
(
a
) =
α
(0 ∗ (∗0 ∗
a
)) = 0 ∗ (∗0 ∗
α
(
a
)). Thus
α
(
a
) ∈
L_{p}
(
X
). Similarly,
β
(
a
) ∈
L_{p}
(
X
).
(3) Let
a
∈
L_{p}
(
X
): Using (2), (a1) and (a14), we have
(4) Let
a
,
b
∈
L_{p}
(
X
): Then
a
+
b
∈
L_{p}
(
X
): Using (3), we have
This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.24
(
[5]
).
Let d
_{(α, β)}
,
be a regular
(
α
,
β
)
derivation of a BCIalgebra X
.
Then

(1) (∀a∈X)(a∈Lp(X) ⇒d(α,β)(a) ∈Lp(X)).

(2) (∀a∈X) (a∈Lp(X) ⇒α(a),β(a) ∈Lp(X)).

(3) (∀a∈Lp(X))(d(α,β)(a) =d(α,β)(0) +α(a)).

(4) (∀a,b∈Lp(X))(d(α,β)(a+b) =d(α,β)(a) +d(α,β)(b) −d(α,β)(0)).
Theorem 3.25.
Let X be a torsion free BCIalgebra and F be a regular generalized
(
α
,
β
)
derivation on X such that α
◦
F
=
F
.
If F
^{2}
= 0
on Lp
(
X
),
then F
= 0
on Lp
(
X
).
Proof
. Let us suppose
F
^{2}
= 0 on
Lp
(
X
). If
x
∈
Lp
(
X
), then
x
+
x
∈
Lp
(
X
) and so by using Theorem 3.23 (3) and (4), we have
Since
X
is a torsion free, therefore
F
(
x
) = 0 for all
x
∈
X
implying thereby
F
= 0. This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.26
(
[5]
).
Let X be a torsion free BCIalgebra and d
_{(α, β)}
be a regular
(
α
,
β
)
derivation on X such that α
◦
d
_{(α, β)}
=
d
_{(α, β)}
.
If
on Lp
(
X
),
then d
_{(α, β)}
= 0
on Lp
(
X
).
Theorem 3.27.
Let X be a torsion free BCIalgebra and F, F′ be two regular generalized
(
α
,
β
)
derivations on X such that α
◦
F′
=
F′
.
If F
◦
F′
= 0
on Lp
(
X
),
then F′
= 0
on Lp
(
X
).
Proof
. Let us suppose
F
◦
F′
= 0 on
Lp
(
X
). If
x
∈
Lp
(
X
), then
x
+
x
∈
Lp
(
X
) and so by using Theorem 3.23 (1) and (2), we have
Since
X
is a torsion free, therefore
F′
(
x
) = 0 for all
x
∈
X
and so
F′
= 0. This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.28
(
[5]
).
Let X be a torsion free BCIalgebra and d
_{(α, β)}
,
be two regular
(
α
,
β
)
derivations on X such that
If
= 0
on Lp
(
X
),
then
on Lp
(
X
).
Proposition 3.29.
Let F be a regular generalized
(
α
,
β
)
derivation of a BCIalgebra X
.
If F
^{2}
= 0
on L_{p}
(
X
),
then
for all x
∈
L_{p}
(
X
).
Proof
. Assume that
F
^{2}
= 0 on
L_{p}
(
X
): If
x
∈
L_{p}
(
X
), then
x
+
x
∈
L_{p}
(
X
) and so by using Theorem 3.23 (3) and (4), we have
Hence
for all
x
∈
L_{p}
(
X
).
This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.30
(
[5]
).
Let d
_{(α, β)}
be a regular
(
α
,
β
)
derivation of a BCIalgebra X
.
If
on L_{p}
(
X
),
then
for all x
∈
L_{p}
(
X
).
Proposition 3.31.
Let F and F′ be two regular generalized
(
α
,
β
)
derivations of a BCIalgebra X
.
If F
◦
F′
= 0
on L_{p}
(
X
),
then
for all x
∈
L_{p}
(
X
).
Proof
. Let
x
∈
L_{p}
(
X
). Then
x
+
x
∈
L_{p}
(
X
), and so
F′
(
x
+
x
) ∈
L_{p}
(
X
) by Theorem 3.23 (1). It follows from Theorem 3.23 (3) and (4) that
so that
for all
x
∈
L_{p}
(
X
).
This completes the proof.
If we take
F
=
d
_{(α, β)}
, then we have the following corollary.
Corollary 3.32
(
[5]
).
Let d
_{(α, β)}
and
be two regular
(
α
,
β
)
derivations of a BCIalgebra X
.
If
on L_{p}
(
X
),
then
for all x
∈
L_{p}
(
X
).
BIO
Abdullah M. Alroqi received M.Sc. from University of Missouri, Kansas city, United States and Ph.D at School of Mathematics and Statistics, University of Birmingham, United Kingdom. His mathematical research areas are Algebras related to logic, Finite Group Theory, Soluble groups, Classification of finite simple groups and Representation Theory.
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.
email: aalroqi@kau.edu.sa
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