# Ont-Derivations of BCI-Algebras

Authors: G. Muhiuddin; Abdullah M. Al-roqi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce the notion oft-derivation of a BCI-algebra and investigate related properties. Moreover, we study t-derivations in a p-semisimple BCI-algebra and establish some results on t-derivations in a p-semisimple BCI-algebra.


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## Body

## 1. Introduction

The notion of BCK-algebra was proposed by Imai and Iséki in 1966 [1]. In the same year,  Iséki introduced the notion of a BCI-algebra [2], which is a generalization of a BCK-algebra. A series of interesting notions concerning BCI-algebras were introduced and studied, several papers have been written on various aspects of these algebras [3–5]. Recently, in the year 2004 [6], Jun and Xin have applied the notion of derivation in BCI-algebras which is defined in a way similar to the notion of derivation in rings and near-rings theory which was introduced by Posner in 1957 [7]. In fact, the notion of derivation in ring theory is quite old and plays a significant role in analysis, algebraic geometry and algebra.After the work of Jun and Xin (2004) [6], many research articles have appeared on the derivations of BCI-algebras in different aspects as follows: in 2005 [8],   Zhan and Liu have given the notion of f-derivation of BCI-algebras and studied p-semisimple BCI-algebras by using the idea of regular f-derivation in BCI-algebras. In 2006 [9], Abujabal and  Al-Shehri have extended the results of BCI-algebras. Further, in the next year 2007 [10], they defined and studied the notion of left derivation of BCI-algebras and investigated some properties of left derivation in p-semisimple BCI-algebras. In 2009 [11], Öztürk and Çeven have defined the notion of derivation and generalized derivation determined by a derivation for a complicated subtraction algebra and discussed some related properties. Also, in 2009 [12], Öztürk et al. have introduced the notion of generalized derivation in BCI-algebras and established some results. Further, they have given the idea of torsion free BCI-algebra and explored some properties. In 2010 [13], Al-Shehri has applied the notion of left-right (resp., right-left) derivation in BCI-algebra to B-algebra and obtained some of its properties. In 2011 [14],   Ilbira et al. have studied the notion of left-right (resp., right-left) symmetric biderivation in BCI-algebras.Motivated by a lot of work done on derivations of BCI-algebras and on derivations of other related abstract algebraic structures, in this paper we introduce the notion oft-derivations on BCI-algebras and obtain some of its related properties. Further, we characterize the notion of p-semisimple BCI-algebra X by using the notion of t-derivation and show that if dt and dt′ are t-derivations on X, then dt∘dt′ is also a t-derivation and dt∘dt′=dt′∘dt. Finally, we prove that dt*dt′=dt′*dt, where dt and dt′ are t-derivations on a p-semisimple BCI-algebra.


## 2. Preliminaries

We review some definitions and properties that will be useful in our results.Definition 2.1 (see [2]).
LetX be a set with a binary operation “*” and a constant 0. Then (X,*,0) is called a BCI algebra if the following axioms are satisfied for all x,y,z∈X:  (i)
((x*y)*(x*z))*(z*y)=0,(ii)
(x*(x*y))*y=0,(iii)
x*x=0,(iv)
x*y=0 and y*x=0⇒x=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 BCI-algebra X satisfying 0≤x for all x∈X, is called BCK-algebra (see [1]).
In any BCI-algebraX for all x,y∈X, the following properties hold. (1)
(x*y)*z=(x*z)*y.(2)
x*0=x.(3)
(x*z)*(y*z)≤x*y.(4)
x*0=0 implies x=0.(5)
x≤y⇔x*z≤y*z and z*y≤z*x. A BCI-algebra X is said to be associative if for all x,y,z∈X, the following holds:(6)
(x*y)*z=x*(y*z) [4]. Let X be a BCI-algebra, we denote X+={x∈X∣0≤x}, the BCK-part of X and by G(X)={x∈X∣0*x=x}, the BCI-G part of X. If X+={0}, then X is called a p-semisimple BCI-algebra. In a p-semisimple BCI-algebra X, the following properties hold.(7)
x*(x*y)=y.(8)
x*(0*y)=y*(0*x).(9)
x*y=0 implies x=y.(10)
(x*z)*(y*z)=x*y.(11)
x*a=x*b implies a=b that is left cancelable.(12)
a*x=b*x implies a=b that is right cancelable.Definition 2.2 (see [6]).
A subsetS of a BCI-algebra X is called subalgebra of X if x*y∈S whenever x,y∈S.
For a BCI-algebraX, we denote x∧y=y*(y*x) for all x,y∈X [6]. For more details we refer to [3, 5, 6].


## 3.t-Derivations in a BCI-Algebra/p-Semisimple BCI-Algebra

The following definitions introduce the notion oft-derivation for a BCI-algebra.Definition 3.1.
LetX be a-BCI-algebra. Then for any t∈X, we define a self map dt:X→X by dt(x)=x*t for all x∈X.Definition 3.2.
LetX be a BCI-algebra. Then for any t∈X, a self map dt:X→X is called a left-right t-derivation or (l,r)-t-derivation of X if it satisfies the identity dt(x*y)=(dt(x)*y)∧(x*dt(y)) for all x,y∈X.
Similarly, we get the following.Definition 3.3.
LetX be a BCI-algebra. Then for any t∈X, a self map dt:X→X is called a right-left t-derivation or (r,l)-t-derivation of X if it satisfies the identity dt(x*y)=(x*dt(y))∧(dt(x)*y) for all x,y∈X.
Moreover, ifdt is both a (l,r)- and a (r,l)-t-derivation on X, we say that dt is a t-derivation on X.Example 3.4.
LetX={0,1,2} be a BCI-algebra with the following Cayley table:
(3.1)*012000211022220
For any t∈X, define a self map dt:X→X by dt(x)=x*t for all x∈X. Then it is easily checked that dt is a t-derivation of  X.Proposition 3.5.
Letdt be a self map of an associative BCI-algebra X. Then  dt is a (l,r)-t-derivation of X.Proof.
LetX be an associative BCI-algebra, then we have
(3.2)dt(x*y)=(x*y)*t={x*(y*t)}*0by  Property(6)and  (2)={x*(y*t)}*[{x*(y*t)}*{x*(y*t)}]by  Property(iii)={x*(y*t)}*[{x*(y*t)}*{(x*y)*t}]by  Property(6)={x*(y*t)}*[{x*(y*t)}*{(x*t)*y}]by  Property(1)=((x*t)*y)∧(x*(y*t))=(dt(x)*y)∧(x*dt(y)).Proposition 3.6.
Letdt be a self map of an associative BCI-algebra X. Then, dt is a (r,l)-t-derivation of X.Proof.
LetX be an associative BCI-algebra, then we have
(3.3)dt(x*y)=(x*y)*t={(x*t)*y}*0by  Property(1)and(2)={(x*t)*y}*[{(x*t)*y}*{(x*t)*y}](asx*x=0)={(x*t)*y}*[{(x*t)*y}*{(x*y)*t}]by  Property(1)={(x*t)*y}*[{(x*t)*y}*{x*(y*t)}]by  Property(6)=(x*(y*t))∧((x*t)*y)(asy*(y*x)=x∧y)=(x*dt(y))∧(dt(x)*y).
Combining Propositions 3.5 and 3.6, we get the following Theorem.Theorem 3.7.
Letdt be a self map of an associative BCI-algebra X. Then, dt is a t-derivation of X.Definition 3.8.
A self mapdt of a BCI-algebra X is said to be t-regular if dt(0)=0.Example 3.9.
LetX={0,a,b} be a BCI-algebra with the following Cayley table:
(3.4)*0ab000baa0bbbb0
(i) For anyt∈X, define a self map dt:X→X by
(3.5)dt(x)=x*t={bifx=0,a0ifx=b.
Then it is easily checked that dt is (l,r) and  (r,l)-t-derivations of X, which is not t-regular.
(ii) For anyt∈X, define a self map dt′:X→X by
(3.6)dt′(x)=x*t={0ifx=0,abifx=b.
Then it is easily checked that dt′ is (l,r) and  (r,l)-t-derivations of X, which is t-regular.Proposition 3.10.
Letdt be a self map of a BCI-algebra X. Then  (i)
Ifdt is a (l,r)-t-derivation of X, then dt(x)=dt(x)∧x for all x∈X.(ii)
Ifdt is a (r,l)-t-derivation of X, then dt(x)=x∧dt(x) for all x∈X if and only if dt is t-regular.Proof of (i).
Letdt be a (l,r)-t-derivation of X, then
(3.7)dt(x)=dt(x*0)=(dt(x)*0)∧(x*dt(0))=dt(x)∧{x*dt(0)}={x*dt(0)}*[{x*dt(0)}*dt(x)]={x*dt(0)}*[{x*dt(x)}*dt(0)]≤x*{x*dt(x)}byProperty(3)=dt(x)∧x.
But dt(x)∧x≤dt(x) is trivial so (i) holds.Proof of (ii).
Letdt be a (r,l)-t-derivation of X. If dt(x)=x∧dt(x) then
(3.8)dt(0)=0∧dt(0)=dt(0)*{dt(0)*0}=dt(0)*dt(0)=0
thereby implying dt is t-regular. Conversely, suppose that dt is t-regular, that is  dt(0)=0, then we have
(3.9)dt(x)=dt(x*0)=(x*dt(0))∧(dt(x)*0)=(x*0)∧dt(x)=x∧dt(x).
This completes the proof.Theorem 3.11.
Letdt be a (l,r)-t-derivation of a p-semisimple BCI-algebra X. Then the following hold:  (i)
dt(0)=dt(x)*x for all x∈X.(ii)
dt is one-one.(iii)
Ifdt is t-regular, then it is an identity map.(iv)
if there is an elementx∈X such that dt(x)=x, then dt is identity map.(v)
ifx≤y, then dt(x)≤dt(y) for all x,y∈X.Proof of (i).
Letdt be a (l,r)-t-derivation of a p-semisimple BCI-algebra X. Then for all x∈X, we have x*x=0 and so (3.10)dt(0)=dt(x*x)=(dt(x)*x)∧(x*dt(x))={x*dt(x)}*[{x*dt(x)}*{dt(x)*x}]=dt(x)*xby  property(7).Proof of (ii).
Letdt(x)=dt(y)⇒x*t=y*t, then by property (12), we have x=y and so dt is one-one.Proof of (iii).
Letdt be t-regular and x∈X. Then, 0=dt(0) so by the above part (i), we have 0=dt(x)*x and hence by property (9), we obtain dt(x)=x for all x∈X. Therefore, dt is the identity map.Proof of (iv).
It is trivial and follows from the above part (iii).Proof of (v).
Letx≤y implying x*y=0. Now,
(3.11)dt(x)*dt(y)=(x*t)*(y*t)=x*yby  property(10)=0.
Therefore,  dt(x)≤dt(y). This completes the proof.Definition 3.12.
Letdt be a t-derivation of a BCI-algebra X. Then, dt is said to be an isotone t-derivation if x≤y⇒dt(x)≤dt(y) for all  x,y∈X.Example 3.13.
In Example3.9(ii), dt′ is an isotone t-derivation, while in Example 3.9(i), dt is not an isotone t-derivation.Proposition 3.14.
LetX be a BCI-algebra and dt be a t-derivation on X. Then for all x,y∈X, the following hold:  (i)
Ifdt(x∧y)=dt(x)∧dt(y), then dt is an isotone t-derivation.(ii)
Ifdt(x*y)=dt(x)*dt(y), then dt is an isotone t-derivation.Proof of (i).
Letdt(x∧y)=dt(x)∧dt(y). If x≤y⇒x∧y=x for all x,y∈X. Therefore, we have
(3.12)dt(x)=dt(x∧y)=dt(x)∧dt(y)≤dt(y).
Henceforth dt(x)≤dt(y) which implies that dt is an isotone t-derivation.Proof of (ii).
Letdt(x*y)=dt(x)*dt(y). If x≤y⇒x*y=0 for all x,y∈X. Therefore, we have
(3.13)dt(x)=dt(x*0)=dt{x*(x*y)}=dt(x)*dt(x*y)=dt(x)*{dt(x)*dt(y)}≤dt(y)by  property(ii).
Thus,  dt(x)≤dt(y). This completes the proof.Theorem 3.15.
Letdt be a t-regular (r,l)-t-derivation of a BCI-algebra X. Then, the following hold:  (i)
dt(x)≤x for all x∈X.(ii)
dt(x)*y≤x*dt(y) for all x,y∈X.(iii)
dt(x*y)=dt(x)*y≤dt(x)*dt(y) for all x,y∈X.(iv)
ker(dt):={x∈X:dt(x)=0} is a subalgebra of X.Proof of (i).
For anyx∈X, we have dt(x)=dt(x*0)=(x*dt(0))∧(dt(x)*0)=(x*0)∧(dt(x)*0)=x∧dt(x)≤x.Proof of (ii).
Sincedt(x)≤x for all x∈X, then dt(x)*y≤x*y≤x*dt(y) and hence the proof follows.Proof of (iii).
For anyx,y∈X, we have
(3.14)dt(x*y)=(x*dt(y))∧(dt(x)*y)={dt(x)*y}*[{dt(x)*y}*{x*dt(y)}]={dt(x)*y}*0=dt(x)*y≤dt(x)*dt(y).Proof of (iv).
Letx,y∈ker(dt)⇒dt(x)=0=dt(y). From (iii), we have dt(x*y)≤dt(x)*dt(y)=0*0=0 implying dt(x*y)≤0 and so dt(x*y)=0. Therefore, x*y∈ker(dt). Consequently ker(dt) is a subalgebra of X. This completes the proof.Definition 3.16.
LetX be a BCI-algebra and let dt, dt′ be two self  maps of X. Then we define dt∘dt′:X→X by (dt∘dt′)(x)=dt(dt′(x)) for all x∈X.Example 3.17.
LetX={0,a,b} be a BCI algebra which is given in Example 3.4. Let dt and dt′ be two self maps on X as defined in Example 3.9(i) and  Example 3.9(ii), respectively.
Now, define a self mapdt∘dt′:X→X by
(3.15)(dt∘dt′)(x)={0ifx=a,bbifx=0.
Then, it is easily checked that (dt∘dt′)(x)=dt(dt′(x)) for all x∈X.Proposition 3.18.
LetX be a p-semisimple BCI-algebra X and let dt, dt′ be (l,r)-t-derivations of X. Then,  dt∘dt′ is also a (l,r)-t-derivation of X.Proof.
LetX be a p-semisimple BCI-algebra. dt and dt′ are (l,r)-t-derivations of X. Then for all x,y∈X, we get
(3.16)(dt∘dt′)(x*y)=dt(dt′(x*y))=dt[(dt′(x)*y)∧(x*dt′(y))]=dt[(x*dt′(y))*{(x*dt′(y))*(dt′(x)*y)}]=dt(dt′(x)*y)by  property(7)={x*dt(dt′(y))}*[{x*dt(dt′(y))}*{dt(dt′(x)*y)}]={dt(dt′(x)*y)}∧{x*dt(dt′(y))}=((dt∘dt′)(x)*y)∧(x*(dt∘dt′)(y)).
Therefore, (dt∘dt′) is a (l,r)-t-derivation of X.
Similarly, we can prove the following.Proposition 3.19.
LetX be a p-semisimple BCI-algebra and let dt, dt′ be (r,l)-t-derivations of X. Then dt∘dt′ is also a (r,l)-t-derivation of X.Combining Propositions3.18 and  3.19, we get the following.Theorem 3.20.
LetX be a p-semisimple BCI-algebra and let dt, dt′ be  t-derivations of X. Then,  dt∘dt′ is also a t-derivation of X.Now, we prove the following theorem.Theorem 3.21.
LetX be a p-semisimple BCI-algebra and let dt, dt′ be t-derivations of X. Then dt∘dt′=dt′∘dt.Proof.
LetX be a p-semisimple BCI-algebra. dt and dt′, t-derivations of X. Suppose dt′ is a (l,r)-t-derivation, then for all x,y∈X, we have
(3.17)(dt∘dt′)(x*y)=dt(dt′(x*y))=dt[(dt′(x)*y)∧(x*dt′(y))]=dt[(x*dt′(y))*{(x*dt′(y))*(dt′(x)*y)}]=dt(dt′(x)*y)by  property(7).
As dt is a (r,l)-t-derivation, then
(3.18)=(dt′(x)*dt(y))∧(dt(dt′(x))*y)=dt′(x)*dt(y).
Again, if dt is a (r,l)-t-derivation, then we have
(3.19)(dt′∘dt)(x*y)=dt′[dt(x*y)]=dt′[(x*dt(y))∧(dt(x)*y)]=dt′[x*dt(y)]by  property(7)
But dt′ is a (l,r)-t-derivation, then
(3.20)=(dt′(x)*dt(y))∧(x*dt′(dt(y)))=dt′(x)*dt(y).
Therefore from (3.18) and (3.20), we obtain
(3.21)(dt∘dt′)(x*y)=(dt′∘dt)(x*y).
By putting y=0, we get
(3.22)(dt∘dt′)(x)=(dt′∘dt)(x)∀x∈X.
Hence, dt∘dt′=dt′∘dt. This completes the proof.Definition 3.22.
LetX be a BCI-algebra and let dt, dt′ be two self  maps of X. Then we define dt*dt′:X→X by (dt*dt′)(x)=dt(x)*dt′(x) for all x∈X.Example 3.23.
LetX={0,a,b} be a BCI algebra which is given in Example 3.4. Let dt and dt′ be two self  maps on X as defined in Example 3.9(i) and Example 3.9(ii), respectively.
Now, define a self mapdt*dt′:X→X by
(3.23)(dt*dt′)(x)={0ifx=a,bbifx=0.
Then, it is easily checked that (dt*dt′)(x)=dt(x)*dt′(x) for all x∈X.Theorem 3.24.
LetX be a p-semisimple BCI-algebra and let dt, dt′ be t-derivations of X. Then dt*dt′=dt′*dt.Proof.
LetX be a p-semisimple BCI-algebra. dt and dt′, t-derivations of X.
Sincedt′ is a (r,l)-t-derivation of X, then for all x,y∈X, we have
(3.24)(dt∘dt′)(x*y)=dt[dt′(x*y)]=dt[(x*dt′(y))∧(dt′(x)*y)]=dt[x*dt′(y)]by  property(7).
But dt is a (l,r)-t-derivation, so
(3.25)=(dt(x)*dt′(y))∧(x*dt(dt′(y)))=dt(x)*dt′(y).
Again, if dt′ is a (l,r)-t-derivation of X, then for all x,y∈X, we have
(3.26)(dt∘dt′)(x*y)=dt[dt′(x*y)]=dt[(dt′(x)*y)∧(x*dt′(y))]=dt[(x*dt′(y))*{(x*dt′(y))*(dt′(x)*y)}]=dt(dt′(x)*y)by  property(7).
As dt is a (r,l)-t-derivation, then
(3.27)=(dt′(x)*dt(y))∧(dt(dt′(x))*y)=dt′(x)*dt(y).
Henceforth from (3.25) and (3.27), we conclude
(3.28)dt(x)*dt′(y)=dt′(x)*dt(y)
By putting y=x, we get
(3.29)dt(x)*dt′(x)=dt′(x)*dt(x)(dt*dt′)(x)=(dt′*dt)(x)∀x∈X.
Hence,  dt*dt′=dt′*dt. This completes the proof.


## 4. Conclusion

Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. The theory of derivations of algebraic structures is a direct descendant of the development of classical Galois theory (namely, Suzuki [15] and Van der Put and Singer [16, 17]) and the theory of invariants. An extensive and deep theory has been developed for derivations in algebraic structures viz. BCI-algebras, C*-algebras, commutative Banach algebras and Galois theory of linear differential equations (see, e.g.,  Jun and Xin  [6], Ara and Mathieu [18], Bonsall and Duncan  [19],  Murphy [20] and Villena [21] where further references can be found). It plays a significant role in functional analysis; algebraic geometry; algebra and linear differential equations.In the present paper, we have considered the notion oft-derivations in BCI-algebras and investigated the useful properties of the t-derivations in BCI-algebras. Finally, we investigated the notion of t-derivations in a p-semisimple BCI-algebra and established some results on t-derivations in a p-semisimple BCI-algebra. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras [11], B-algebras [13], MV-algebras [22], d-algebras, Q-algebras and so forth. In future we can study the notion of t-derivations on various algebraic structures which may have a lot of applications in different branches of theoretical physics, engineering and computer science. It is our hope that this work would serve as a foundation for the further study in the theory of derivations of BCK/BCI-algebras.In our future study oft-derivations in BCI-algebras, may be the following topics should be considered:  (1)
to find the generalizedt-derivations of BCI-algebras,(2)
to find more results int-derivations of BCI-algebras and its applications,(3)
to find thet-derivations of B-algebras, Q-algebras, subtraction algebras, d-algebra and so forth.



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