# On Symmetric Left Bi-Derivations inBCI-Algebras

Authors: G. Muhiuddin; Abdullah M. Al-roqi; Young Bae Jun; Yilmaz Ceven
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

The notion ofsymmetric left bi-derivation of a BCI-algebra X is introduced, and related properties are investigated. Some results on componentwise regular and d-regular symmetric left bi-derivations are obtained. Finally, characterizations of a p-semisimple BCI-algebra are explored, and it is proved that, in a p-semisimple BCI-algebra, F is a symmetric left bi-derivation if and only if it is a symmetric bi-derivation.


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## Body

## 1. Introduction

BCK-algebras and BCI-algebras are two classes of nonclassical logic algebras which were introduced by Imai and Iséki in 1966 [1, 2]. They are algebraic formulation of BCK-system and BCI-system in combinatory logic. Later on, the notion of BCI-algebras has been extensively investigated by many researchers (see [3–6], and references therein). The notion of a BCI-algebra generalizes the notion of a BCK-algebra in the sense that every BCK-algebra is a BCI-algebra but not vice versa (see [7]). Hence, most of the algebras related to thet-norm-based logic such as MTL [8], BL, hoop, MV [9] (i.e lattice implication algebra), and Boolean algebras are extensions of BCK-algebras (i.e. they are subclasses of BCK-algebras) which have a lot of applications in computer science (see [10]). This shows that BCK-/BCI-algebras are considerably general structures.Throughout our discussion,Xwill denote a BCI-algebra unless otherwise mentioned. In the year 2004, Jun and Xin [11] 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. Using this concept as defined they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of ap-semisimple BCI-algebra. For a self-mapdof a BCI-algebra, they defined ad-invariant ideal and gave conditions for an ideal to bed-invariant. According to Jun and Xin, a self mapd:X→Xis called a left-right derivation (briefly(l,r)-derivation) ofXifd(x*y)=d(x)*y∧x*d(y)holds for allx,y∈X. Similarly, a self mapd:X→Xis called a right-left derivation (briefly(r,l)-derivation) ofXifd(x*y)=x*d(y)∧d(x)*yholds for allx,y∈X. Moreover, ifdis both(l,r)- and(r,l)-derivation, it is a derivation onX. After the work of Jun and Xin [11], many research articles have appeared on the derivations of BCI-algebras and a greater interest has been devoted to the study of derivations in BCI-algebras on various aspects (see [12–17]).Inspired by the notions ofσ-derivation [18], left derivation [19], and symmetric bi-derivations [20, 21] in rings and near-rings theory, many authors have applied these notions in a similar way to the theory of BCI-algebras (see [12, 13, 17]). For instantce in 2005 [17], Zhan and Liu have given the notion off-derivation of BCI-algebras as follows: a self mapdf:X→Xis said to be a left-rightf-derivation or(l,r)-f-derivation ofXif it satisfies the identitydf(x*y)=df(x)*f(y)∧f(x)*df(y)for allx,y∈X. Similarly, a self mapdf:X→Xis said to be a right-leftf-derivation or(r,l)-f-derivation ofXif it satisfies the identitydf(x*y)=f(x)*df(y)∧df(x)*f(y)for allx,y∈X. Moreover, ifdfis both(l,r)- and(r,l)-f-derivation, it is said thatdfis anf-derivation, wherefis an endomorphism. In the year 2007, Abujabal and Al-Shehri [12] defined and studied the notion of left derivation of BCI-algebras as follows: a self mapD:X→Xis said to be a left derivation ofXif satisfyingD(x*y)=x*D(y)∧y*D(x)for allx,y∈X. Furthermore, in 2011 [13], Ilbira et al. have introduced the notion of symmetric bi-derivations in BCI-algebras. Following [13], a mappingD(·,·):X×X→Xis said to be symmetric ifF(x,y)=F(y,x)holds for all pairsx,y∈X.A symmetric mappingD(·,·):X×X→Xis called left-right symmetric bi-derivation (briefly(l,r)-symmetric bi-derivation) if it satisfies the identityD(x*y,z)=D(x,z)*y∧x*D(y,z)for allx,y,z∈X.Dis called right-left symmetric bi-derivation (briefly(r,l)-symmetric bi-derivation) if it satisfies the identityD(x*y,z)=x*D(y,z)∧D(x,z)*yfor allx,y,z∈X. Moreover, ifDis both a(l,r)- and a(r,l)-symmetric bi-derivation, it is said thatDis a symmetric bi-derivation onX.Motivated by the notion ofsymmetric bi-derivations [13] in the theory of BCI-algebras, in the present analysis, we introduced the notion of symmetric left bi-derivations on BCI-algebras and investigated related properties. Further, we obtain some results on componentwise regular andd-regular symmetric left bi-derivations. Finally, we characterize the notion ofp-semisimple BCI-algebraXby using the concept of symmetric left bi-derivation and show that, in a p-semisimple BCI-algebraX,Fis a symmetric left bi-derivation if and only if it is a symmetric bi-derivation.


## 2. Preliminaries

We begin with the following definitions and properties that will be needed in the sequel.A nonempty setXwith a constant0and a binary operation*is called a BCI-algebra if for allx,y,z∈Xthe following conditions hold: (I)
((x*y)*(x*z))*(z*y)=0,(II)
(x*(x*y))*y=0,(III)
x*x=0,(IV)
x*y=0andy*x=0implyx=y.Define a binary relation≤onXby lettingx*y=0if and only ifx≤y. Then(X,≤)is a partially ordered set. A BCI-algebraXsatisfying0≤xfor allx∈X is called BCK-algebra.ABCI-algebraXhas the following properties for allx,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=0impliesx=0.For aBCI-algebraX, denote byX+(resp., G(X)) theBCK-part (resp., theBCI-G part) ofX; that is,X+is the set of allx∈Xsuch that0≤x(resp., G(X):={x∈X∣0*x=x}). Note thatG(X)∩X+={0}(see [22]). IfX+={0}, thenXis called ap-semisimpleBCI-algebra. In ap-semisimpleBCI-algebraX, 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=0impliesx=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).LetXbe ap-semisimple BCI-algebra. We define addition “+” asx+y=x*(0*y)for allx,y∈X. Then(X,+)is an abelian group with identity0andx-y=x*y. Conversely, let(X,+)be an abelian group with identity0, and letx*y=x-y. ThenXis ap-semisimple BCI-algebra andx+y=x*(0*y)for allx,y∈X(see [6]).For aBCI-algebraX, we denotex∧y=y*(y*x), in particular0*(0*x)=ax, andLp(X):={a∈X∣x*a=0⇒x=a,∀x∈X}.We call the elements ofLp(X)thep-atoms ofX. For anya∈X, letV(a):={x∈X∣a*x=0}, which is called the branch ofXwith respect toa. It follows thatx*y∈V(a*b)wheneverx∈V(a)andy∈V(b)for allx,y∈Xand alla,b∈Lp(X). Note thatLp(X)={x∈X∣ax=x},which is thep-semisimple part ofX, andXis ap-semisimple BCI-algebra if and only ifLp(X)=X(see [23, Proposition 3.2]). Note also thatax∈Lp(X); that is,0*(0*ax)=ax, which implies thatax*y∈Lp(X)for ally∈X. It is clear thatG(X)⊂Lp(X), andx*(x*a)=aanda*x∈Lp(X)for alla∈Lp(X)and allx∈X. LetD(·,·):X×X→Xbe a symmetric mapping. Then for allx∈X, a mappingd:X→Xdefined byd(x)=D(x,x)is called trace ofD[13]. For more details, refer to [3, 4, 6, 11, 22, 23].


## 3. Symmetric Left Bi-Derivations

The following definition introduces the notion ofsymmetric left bi-derivation for a BCI-algebraX.Definition 1.
A symmetric mappingF(·,·):X×X→Xis called a symmetric left bi-derivation ofXif it satisfies the following identity:
(1)(∀x,y,z∈X)(F(x*y,z)=(x*F(y,z))∧(y*F(x,z))).Example 2 (see [24]).
Consider ap-semisimple BCI-algebraX={0,3,4,5}with the following Cayley table:
(2)*034500345330544450355430
Define a mappingF(·,·):X×X→Xby
(3)F(0,0)=F(3,3)=F(4,4)=F(5,5)=0,F(0,3)=F(3,0)=3,F(0,4)=F(4,0)=4,F(0,5)=F(5,0)=5,F(3,4)=F(4,3)=5,F(3,5)=F(5,3)=4,F(4,5)=F(5,4)=3.
It is routine to verify thatFis a symmetric left bi-derivation ofX.Theorem 3.
LetF(·,·):X×X→Xbe a symmetric left bi-derivation ofX.Then (1)
(∀z∈X)(a∈G(X)⇒F(a,z)∈G(X)).(2)
(∀z∈X)(a∈Lp(X)⇒F(a,z)∈Lp(X)).(3)
(∀z∈X)(a∈Lp(X)⇒F(a,z)=0+F(a,z)).(4)
(∀z∈X)(a∈Lp(X)⇒F(a,z)=a*F(0,z)=a+F(0,z)).Proof.
(1) Leta∈G(X).Then0*a=a, and so
(4)F(a,z)=F(0*a,z)=(0*F(a,z))∧(a*F(0,z))=(a*F(0,z))*((a*F(0,z))*(0*F(a,z)))=0*F(a,z),
since0*F(a,z)∈Lp(X).HenceF(a,z)∈G(X).
(2) For anya∈Lp(X)impliesa=0*(0*a)and so
(5)F(a,z)=F(0*(0*a),z)=(0*F(0*a,z))∧((0*a)*F(0,z))=((0*a)*F(0,z))*(((0*a)*F(0,z))*(0*F(0*a,z)))=0*F(0*a,z)∈Lp(X).
(3) By (2), we haveF(a,z)∈Lp(X). Then
(6)F(a,z)=0*(0*F(a,z))=0+F(a,z).
(4) For anya∈Lp(X)andz∈X, we have
(7)F(a,z)=F(a*0,z)=(a*F(0,z))∧(0*F(a,z))=(0*F(a,z))*((0*F(a,z))*(a*F(0,z)))=a*F(0,z)=a*(0*F(0,z))=a+F(0,z).
This completes the proof.Using Theorem3, we have the following corollary.Corollary 4.
LetF(·,·):X×X→Xbe a symmetric left bi-derivation andd:X→Xbe the trace ofF.Then (1)
(∀a∈G(X))(d(a)∈G(X)).(2)
(∀a∈Lp(X))(d(a)∈Lp(X)).Theorem 5.
LetFbe a symmetric left bi-derivation ofX.Then (1)
(∀z∈X)(a,b∈Lp(X)⇒F(a+b,z)=a+F(b,z)).(2)
(∀z∈X)(a∈Lp(X)⇒F(a,z)=aifandonlyifF(0,z)=0).(3)
(∀x,y,z∈X)(F(x*y,z)≤x*F(y,z)).(4)
(∀x,y,z∈X)(x*F(x,z)=y*F(y,z)).Proof.
(1) Leta,b∈Lp(X). Then
(8)F(a+b,z)=F(a*(0*b),z)=a*F(0*b,z)∧(0*b)*F(a,z)=a*F(0*b,z)=a*(0*F(b,z)∧b*F(0,z))=a*(0*F(b,z))=a+F(b,z).
(2) SupposeF(a,z)=afor alla∈Lp(X),z∈X. It is clear that, for0∈Lp(X), we haveF(0,z)=0.Conversely let us assume thatF(0,z)=0;then by using Theorem 3(4), we haveF(a,z)=a+F(0,z)=a+0=a.
(3) For anyx,y,z∈X, we have
(9)F(x*y,z)=(x*F(y,z))∧(y*F(x,z))=(y*F(x,z))*((y*F(x,z))*(x*F(y,z)))≤x*F(y,z).
(4) For anyx,z∈X, we have
(10)F(0,z)=F(x*x,z)=(x*F(x,z))∧(x*F(x,z))=x*F(x,z).
Thus, we can writeF(0,z)=x*F(x,z)=y*F(y,z)for anyy∈X.This completes the proof.Definition 6.
Asymmetric left bi-derivationF(·,·):X×X→Xof a BCI-algebra X is said to be componentwise regular if F(0,z)=0 for all z∈X. In particular, F is called d-regular if F(0,0)=d(0)=0.Theorem 7.
LetFbe a symmetric left bi-derivation of BCI-algebraX. ThenXis a BCK-algebra if and only ifFis componentwise regular symmetric left bi-derivation.Proof.
SupposeXis a BCK-algebra. Then for anyx,z∈X, we have
(11)F(0,z)=F(0*x,z)=(0*F(x,z))∧(x*F(0,z))=0∧(x*F(0,z))=0.
HenceFis componentwise regular.
Conversely, letFbe a componentwise regular symmetric left bi-derivation. Let for anya∈Lp(X)be such thata≠0. Then
(12)F(a*0,0)=F(a,0)=0.
But it is clear that
(13)a*F(0,0)∧0*F(a,0)=a*0∧0*0=a∧0=0*(0*a)=a≠0,
which is not possible asFis a componentwise regular symmetric left bi-derivation. Thus0is the uniquep-atom. Assume that for somem∈X, we have0*m≠0, thena0*m=0*(0*(0*m))=0, so 0*m∈Lp(X), which is a contradiction. Henceforth, for allm∈X, we have0*m=0implying thereby,Xis a BCK-algebra.
This completes the proof.Theorem 8.
LetFbe a componentwise regular symmetric left bi-derivation of a BCI-algebraX.Then (1)
BothxandF(x,z)belong to the same branch for allx,z∈X.(2)
(∀x,z∈X)(F(x,z)≤x).(3)
(∀x,y,z∈X)(F(x,z)*y≤x*F(y,z)).Proof.
(1) For anyx,z∈X, we get
(14)0=F(0,z)=F(ax*x,z)=(ax*F(x,z))∧(x*F(ax,z))=(x*F(ax,z))*((x*F(ax,z))*(ax*F(x,z)))=ax*F(x,z),
sinceax*F(x,z)∈Lp(X).Henceax≤F(x,z),and soF(x,z)∈V(ax).Obviously,x∈V(ax).
(2) SinceFis componentwise regular,F(0,z)=0.Then
(15)F(x,z)=F(x*0,z)=(x*F(0,z))∧(0*F(x,z))=(x*0)∧(0*F(x,z))=(0*F(x,z))*((0*F(x,z))*x)≤x.
(3) SinceF(x,z)≤xfor allx,z∈Xby (2), using (a3) we obtain
(16)F(x,z)*y≤x*y≤x*F(y,z).
This completes the proof.Next, we prove some results in ap-semisimple BCI-algebra.Theorem 9.
LetFbe a symmetric left bi-derivation of ap-semisimple BCI-algebraX; one has the following assertions.(1)
(∀x,y,z∈X)(F(x*y,z)=x*F(y,z)).(2)
(∀x,y,z∈X)(F(x,z)*x=F(y,z)*y).(3)
(∀x,y,z∈X)(F(x,z)*x=y*F(y,z)).Proof.
(1) LetXbe ap-semisimple BCI-algebra. Then for anyx,y,z∈X, we have
(17)F(x*y,z)=(x*F(y,z))∧(y*F(x,z))=x*F(y,z).
(2) Letx,y,z∈X. Using (I), we have
(18)(x*y)*(x*F(y,z))≤F(y,z)*y,(y*x)*(y*F(x,z))≤F(x,z)*x.
These above inequalities can be rewritten as
(19)((x*y)*(x*F(y,z)))*(F(y,z)*y)=0,((y*x)*(y*F(x,z)))*(F(x,z)*x)=0.
Consequently, we get
(20)((x*y)*(x*F(y,z)))*(F(y,z)*y)=((y*x)*(y*F(x,z)))*(F(x,z)*x)
Also, using Theorem 5(4) and (1), we obtain
(21)(x*y)*F(x*y,z)=(y*x)*F(y*x,z)⟹(x*y)*(x*F(y,z))=(y*x)*(y*F(x,z)).
SinceXis ap-semisimple BCI-algebra, hence, by using (21) and (a12), the above (20) yieldsF(x,z)*x=F(y,z)*y.
(3) We haveF(0,z)=x*F(x,z)by Theorem 5(4). Further, on lettingx=0, we get that F(0,z)*0=F(y,z)*yimpliesF(0,z)=F(y,z)*y. HenceforthF(y,z)*y=x*F(x,z), which amounts to say thatF(x,z)*x=y*F(y,z).
This completes the proof.Theorem 10.
LetXbe ap-semisimple BCI-algebra. ThenFis a symmetric left bi-derivation if and only if it is a symmetric bi-derivation onX.Proof.
Suppose thatFis a symmetric left bi-derivation onX.First, we show thatFis a (r,l)-symmetric bi-derivation onX. Letx,y,z∈X. Using Theorem 9(1) and (a14), we have
(22)F(x*y,z)=x*F(y,z)=(F(x,z)*y)*((F(x,z)*y)*(x*F(y,z)))=(x*F(y,z))∧(F(x,z)*y).
HenceFis a(r,l)-symmetric bi-derivation onX.
Again, we show thatFis a (l,r)-symmetric bi-derivation onX. Letx,y,z∈X. Using Theorem 9(1), (3) and (a15), we have
(23)F(x*y,z)=x*F(y,z)=(x*0)*F(y,z)=(x*(F(0,z)*F(0,z)))*F(y,z)=(x*((x*F(x,z))*(F(y,z)*y)))*F(y,z)=(x*F(y,z))*((x*F(x,z))*(F(y,z)*y))=(x*F(y,z))*((x*F(y,z))*(F(x,z)*y))=(F(x,z)*y)∧(x*F(y,z)).
Conversely, suppose thatFis a symmetric bi-derivation ofX.AsFis a (r,l)-symmetric bi-derivation onX, then for anyx,y,z∈Xand using (a14), we have
(24)F(x*y,z)=(x*F(y,z))∧(F(x,z)*y)=(F(x,z)*y)*((F(x,z)*y)*(x*F(y,z)))=x*F(y,z)=(y*F(x,z))*((y*F(x,z))*(x*F(y,z)))=(x*F(y,z))∧(y*F(x,z)).
HenceFis a symmetric left bi-derivation. This completes the proof.



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