We introduce the notion of t-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.

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 of t-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 [<xref ref-type="bibr" rid="B15">2</xref>]).

Let X 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:

((x*y)*(x*z))*(z*y)=0,

(x*(x*y))*y=0,

x*x=0,

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-algebra X for all x,y∈X, the following properties hold.

(x*y)*z=(x*z)*y.

x*0=x.

(x*z)*(y*z)≤x*y.

x*0=0 implies x=0.

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:

(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.

x*(x*y)=y.

x*(0*y)=y*(0*x).

x*y=0 implies x=y.

(x*z)*(y*z)=x*y.

x*a=x*b implies a=b that is left cancelable.

a*x=b*x implies a=b that is right cancelable.

Definition 2.2 (see [<xref ref-type="bibr" rid="B18">6</xref>]).

A subset S of a BCI-algebra X is called subalgebra of X if x*y∈S whenever x,y∈S.

For a BCI-algebra X, we denote x∧y=y*(y*x) for all x,y∈X [6]. For more details we refer to [3, 5, 6].

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M78"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:math></inline-formula>-Derivations in a BCI-Algebra/<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M79"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>-Semisimple BCI-Algebra

The following definitions introduce the notion of t-derivation for a BCI-algebra.

Definition 3.1.

Let X 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.

Let X 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.

Let X 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, if dt 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.

Let X={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.

Let dt be a self map of an associative BCI-algebra X. Then dt is a (l,r)-t-derivation of X.

Proof.

Let X 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.

Let dt be a self map of an associative BCI-algebra X. Then, dt is a (r,l)-t-derivation of X.

Proof.

Let X 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}](as x*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.

Let dt be a self map of an associative BCI-algebra X. Then, dt is a t-derivation of X.

Definition 3.8.

A self map dt of a BCI-algebra X is said to be t-regular if dt(0)=0.

Example 3.9.

Let X={0,a,b} be a BCI-algebra with the following Cayley table:
(3.4)*0ab000baa0bbbb0

(i) For any t∈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 any t∈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.

Let dt be a self map of a BCI-algebra X. Then

If dt is a (l,r)-t-derivation of X, then dt(x)=dt(x)∧x for all x∈X.

If dt 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).

Let dt 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).

Let dt 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.

Let dt be a (l,r)-t-derivation of a p-semisimple BCI-algebra X. Then the following hold:

dt(0)=dt(x)*x for all x∈X.

dt is one-one.

If dt is t-regular, then it is an identity map.

if there is an element x∈X such that dt(x)=x, then dt is identity map.

if x≤y, then dt(x)≤dt(y) for all x,y∈X.

Proof of (i).

Let dt 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).

Let dt(x)=dt(y)⇒x*t=y*t, then by property (12), we have x=y and so dt is one-one.

Proof of (iii).

Let dt 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).

Let x≤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.

Let dt 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 Example 3.9(ii), dt′ is an isotone t-derivation, while in Example 3.9(i), dt is not an isotone t-derivation.

Proposition 3.14.

Let X be a BCI-algebra and dt be a t-derivation on X. Then for all x,y∈X, the following hold:

If dt(x∧y)=dt(x)∧dt(y), then dt is an isotone t-derivation.

If dt(x*y)=dt(x)*dt(y), then dt is an isotone t-derivation.

Proof of (i).

Let dt(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).

Let dt(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.

Let dt be a t-regular (r,l)-t-derivation of a BCI-algebra X. Then, the following hold:

dt(x)≤x for all x∈X.

dt(x)*y≤x*dt(y) for all x,y∈X.

dt(x*y)=dt(x)*y≤dt(x)*dt(y) for all x,y∈X.

ker(dt):={x∈X:dt(x)=0} is a subalgebra of X.

Proof of (i).

For any x∈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).

Since dt(x)≤x for all x∈X, then dt(x)*y≤x*y≤x*dt(y) and hence the proof follows.

Proof of (iii).

For any x,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).

Let x,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.

Let X 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.

Let X={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 map dt∘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.

Let X 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.

Let X 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.

Let X 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 Propositions 3.18 and 3.19, we get the following.

Theorem 3.20.

Let X 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.

Let X be a p-semisimple BCI-algebra and let dt, dt′ be t-derivations of X. Then dt∘dt′=dt′∘dt.

Proof.

Let X 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.

Let X 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.

Let X={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 map dt*dt′:X→X by
(3.23)(dt*dt′)(x)={0if x=a,bbif x=0.
Then, it is easily checked that (dt*dt′)(x)=dt(x)*dt′(x) for all x∈X.

Theorem 3.24.

Let X be a p-semisimple BCI-algebra and let dt, dt′ be t-derivations of X. Then dt*dt′=dt′*dt.

Proof.

Let X be a p-semisimple BCI-algebra. dt and dt′, t-derivations of X.

Since dt′ 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 of t-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 of t-derivations in BCI-algebras, may be the following topics should be considered:

to find the generalized t-derivations of BCI-algebras,

to find more results in t-derivations of BCI-algebras and its applications,

to find the t-derivations of B-algebras, Q-algebras, subtraction algebras, d-algebra and so forth.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and several useful suggestions. This research is supported by the Deanship of Scientific Research, University of Tabuk, Tabuk, Saudi Arabia.

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