# Generalized Derivations ofBCC-Algebras

Authors: S. M. Bawazeer; N. O. Alshehri; Rawia Saleh Babusail
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

The notion of generalized derivations ofBCC-algebras is introduced, and some related properties are investigated. Also, we consider regular
generalized derivations and the D-invariant on ideals of BCC-algebras. We also
characterized KerD by generalized derivations.


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## Body

## 1. Introduction

Imai and Iséki [1] defined a class of algebras of type (2,0) called BCK-algebras which generalizes on one hand the notion of algebra of sets with the set subtraction as the only fundamental nonnullary operation on the other hand the notion of implication algebra. The class of all BCK-algebras is a quasivariety. Iséki posed an interesting problem (solved by Wroński [2]) whether the class of BCK-algebras is a variety. In connection with this problem, Komori [3] introduced a notion of BCC-algebras, and Dudek [4] redefined the notion of BCC-algebras by using a dual form of the ordinary definition in the sense of Komori. Dudek and Zhang [5] introduced a new notion of ideals in BCC-algebras and described connections between such ideals and congruences. On the other hand, Jun and Xin [6] applied the notion of derivations in ring and near-ring theories to BCI-algebras, and they also introduced a new concept called a regular derivation in BCI-algebras and investigated related properties. They defined a d-derivation ideal and gave conditions for an ideal to be d-derivation. Prabayak and Leerawat [7] introduced the notion of derivation in BCC-algebras. In this paper, the notion of generalized derivations of BCC-algebras is introduced, and some related properties are investigated. Also, we consider regular generalized derivations and the D-invariant on ideals of BCC-algebras. We also characterized KerD by generalized derivations.


## 2. Preliminaries

We review some definitions and properties that will be useful in our results.Definition 1.
LetX be a set with a binary operation “*” and a constant 0. Then (X,*,0) is called a BCC-algebra if the following axioms are satisfied for all x,y,z∈X:(i)
((x*y)*(z*y))*(x*z)=0,(ii)
0*x=0,(iii)
x*0=x,(iv)
x*x=0,(v)
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.In anyBCC-algebra X for all x,y∈X, the following properties hold:(1)
(x*y)*x=0,(2)
x≤y implies x*z≤y*z,(3)
x≤y implies z*y≤z*x.AnyBCK-algebra is a BCC-algebra, but there are BCC-algebras which are not BCK-algebras. Note that a BCC-algebra is a BCK-algebra if and only if satisfies(4)
(x*y)*z=(x*z)*y, or(5)
(x*(x*y))*y=0.Definition 2.
A nonempty subsetS  of a BCC-algebra  X is called subalgebra of  X  if x*y∈S whenever x,y∈S. For a BCC-algebra X, we denote x∧y=y*(y*x) for all x,y∈X.Remark 3.
LetX be a BCC-algebra and x,y∈X; then, consider(1)
0∧x=0,(2)
x∧y≤y.Definition 4.
ABCC-algebra is said to be commutative if and only if satisfies for all x,y∈X,  x*(x*y)=y*(y*x); that is, x∧y=y∧x.Definition 5.
LetX be a BCC-algebra and ϕ≠I⊆X. I is called an ideal of X if it satisfies the following conditions:(i)
0∈I,(ii)
x*y∈I and y∈I  imply x∈I.Definition 6.
LetX be a BCC-algebra. A map d:X→X is a left-right derivation (briefly,(l,r)-derivation) of X, if it satisfies the identity
(1)d(x*y)=(d(x)*y)∧(x*d(y))∀x,y∈X.
Ifd satisfies the identity
(2)d(x*y)=(x*d(y))∧(d(x)*y)∀x,y∈X,
thend is a right-left derivation (briefly, (r,l)-derivation) of X. Moreover, if d is both (l,r)-derivation and (r,l)-derivation, then d is a derivation of X.Definition 7.
A self mapd of a BCC-algebra X is said to be regular if
(3)d(0)=0.Corollary 8.
A derivationd of a BCC-algebra Xis regular.Definition 9.
Letd be a derivation of a BCC-algebra X. An ideal A of X is said to be d-invariant if d(A)⊆A, where d(A)={d(x)∣x∈A}.Proposition 10.
LetX be a BCC-algebra with partial order ≤, and let d be a derivation of X. Then the following hold for all x,y∈X:(i)
d(x)≤x,(ii)
d(x*y)≤d(x)*y,(iii)
d(x*y)≤x*d(y),(iv)
d(d(x))≤x,(v)
d(x*d(x))=0,(vi)
d-1(0)={x∈X:d(x)=0}is a subalgebra ofX.


## 3. Generalized Derivations ofBCC-Algebras

Definition 11.
LetX be a BCC-algebra. A mapping D:X→X  is called a generalized (l,r)-derivation if there exists an (l,r)-derivation d:X→X such that D(x*y)=(D(x)*y)∧(x*d(y)) for all x,y∈X; if there exists an (r,l)-derivation d:X→X such that D(x*y)=(x*D(y))∧(d(x)*y) for all x,y∈X, the mapping D:X→X is called a generalized (r,l)-derivation. Moreover, if D is both a generalized (l,r)- and (r,l)-derivation, we say that D is a generalized derivation.Example 12.
LetX={0,1,2,3} be a BCC-algebra with Cayley table (see Table 1).
Define a mapd:X→X by
(4)d(x)={0ifx=0,1,32ifx=2.
Then d is a derivation of X. Now we define a map D:X→X by
(5)D(x)={0ifx=0,2xifx=1,3.
It is easy to verify that D is a generalized (r,l)-derivation of X.Table 1
*
0
1
2
3
0
0
0
0
0
1
1
0
1
0
2
2
2
0
0
3
3
3
1
0Proposition 13.
A generalized derivationD of a BCC-algebra X is regular.Proof.
Directly from Corollary8, we have that D(0)=0.Proposition 14.
LetD be a self-map of a BCC-algebra X. Then,(i)
ifDis a generalized(l,r)-derivation ofX,thenD(x)=D(x)∧xfor allx∈X;(ii)
ifDis a generalized(r,l)-derivation ofX,thenD(x)=x∧d(x)for allx∈X.Proof.
(i) IfD is a generalized (l,r)-derivation of X, then there exists an (l,r)-derivation d such that D(x*y)=(D(x)*y)∧(x*d(y)) for all x,y∈X. Hence, we get
(6)D(x)=D(x*0)=(D(x)*0)∧(x*d(0))D(x)=D(x)∧(x*d(0))D(x)=(x*d(0))*((x*d(0))*D(x))D(x)=(x*0)*((x*0)*D(x))D(x)=x*(x*D(x))=D(x)∧x.
(ii) IfD is a generalized (r,l)-derivation of X, then there exists an (r,l)-derivation d such that D(x*y)=(x*D(y))∧(d(x)*y) for all x,y∈X. Hence, we get
(7)D(x)=D(x*0)=(x*D(0))∧(d(x)*0)D(x)=(x*0)∧d(x)=x∧d(x).Proposition 15.
LetX be a BCC-algebra with partial order ≤, and let D be a generalized derivation of X. Then the following hold for all x,y∈X:(1)
D(x)≤d(x)≤x,(2)
D(x*y)≤x*d(y),(3)
D(x*y)≤d(x)*y,(4)
D(x*D(x))=0,(5)
D(D(x)*x)=0,(6)
D(d(x)*x)=0,(7)
D(x*d(x))=0,(8)
D(D(x))≤x.Proof.
Consider (1)D(x)=D(x*0)=(x*D(0))∧(d(x)*0)=(x*0)∧d(x)=x∧d(x)≤d(x).
From Proposition10,   d(x)≤x.
(2)D(x*y)=(D(x)*y)∧(x*d(y))≤x*d(y).
(3)D(x*y)=(x*D(y))∧(d(x)*y)≤d(x)*y.
(4)D(x*D(x))=(D(x)*D(x))∧(x*d(D(x)))=0∧(x*d(D(x)))=0.
(5)D(D(x)*x)=(D(x)*D(x))∧(d(D(x))*x)=0∧(d(D(x))*x)=0.
(6)D(d(x)*x)=(D(d(x))*x)∧(d(x)*d(x))=(D(d(x))*x)∧0=0*(0*(D(d(x))*x))=0.
(7)D(x*d(x))=(x*D(d(x)))∧(d(x)*d(x))=(x*D(d(x)))∧0=0*(0*(x*D(d(x))))=0.
(8) From Proposition10, we have
(8)D(D(x))=D(x∧d(x))=D(d(x)*(d(x)*x))=(d(x)*D(d(x)*x))∧d(d(x)*(d(x)*x))=(d(x)*D(0))∧d(d(x)*0)=(d(x)*0)∧d(d(x))=d(x)∧d(d(x))≤d(d(x))≤x.Proposition 16.
LetX be a BCC-algebra. Then Dn(Dn-1(…(D2(D1(x)))…))≤x for n∈N, where D1,D2,…,Dn are generalized derivations of X.Proof.
Trivial.Definition 17.
LetD be a generalized derivation of a BCC-algebra X. An ideal A of X is said to be D-invariant if D(A)⊆A, where
(9)D(A)={D(x)∣x∈A}.Theorem 18.
LetD be a generalized derivation of a BCC-algebra X. Then every ideal A of X is D-invariant.Proof.
LetA be an ideal of a BCC-algebra X. Let y∈D(A). Then y=D(x) for some x∈A. It follows that y*x=D(x)*x=0∈A, which implies that y∈A. Thus D(A)⊆A. Hence, A is D-invariant.Definition 19.
LetX be a BCC-algebra and let D be a generalized derivation. Define a KerD by KerD={x∈X:D(x)=0}.Theorem 20.
LetX be a BCC-algebra and let D be generalized derivation. If y∈Ker(D) and x∈X, then x∧y∈Ker(D).Proof.
Lety∈Ker(D). Then, we get D(y)=0, and so
(10)D(x∧y)=D(y*(y*x))=(D(y)*(y*x))∧(y*d(y*x))=(0*(y*x))∧(y*d(y*x))=0∧(y*d(y*x))=0.
Hence, we havex∧y∈Ker(D).Theorem 21.
LetX be a commutative BCC-algebra and D be a generalized derivation. If x≤y and y∈Ker(D), then x∈Ker(D).Proof.
Letx≤y and y∈Ker(D). Then we get x*y=0 and D(y)=0, and so
(11)D(x)=D(x*0)=D(x*(x*y))=D(y*(y*x))=(D(y)*(y*x))∧(y*d(y*x))=(0*(y*x))∧(y*d(y*x))=0∧(y*d(y*x))=0.
Hence we havex∈Ker(D).Theorem 22.
LetX be a BCC-algebra and let D be a generalized derivation. If x∈Ker(D), one has x*y∈Ker(D) for all y∈X.Proof.
Letx∈Ker(D). Then, D(x)=0. Thus, we have
(12)D(x*y)=(D(x)*y)∧(x*d(y))D(x*y)=(0*y)∧(x*d(y))=0∧(x*d(y))=0.
Hence,x*y∈Ker(D).Theorem 23.
LetX be a BCC-algebra and let D be a generalized derivation.Then Ker(D) is a subalgebra of X.Proof.
Directly from Theorem22.Definition 24.
LetX be a BCC-algebra and let D be a generalized derivation on X. Denote FixD(X)={x∈X:D(x)=x}.Proposition 25.
LetX be a BCC-algebra and let D be a generalized derivation on X. If x∈FixD(X), then d(x)=x.Proof.
Sincex∈FixD(X), then D(x)=x. From Proposition 15, we have D(x)≤d(x)≤x that implies that x≤d(x)≤x. Hence, d(x)=x.Proposition 26.
LetX be a BCC-algebra and let D be a generalized derivation on X. Then FixD(X) is a subalgebra of X.Proof.
Ifx,y∈FixD(X), we get D(x)=x and d(y)=y, and so D(x*y)=(D(x)*y)∧(x*d(y))=(x*y)∧(x*y)=x*y. Hence, x*y∈FixD(X).Proposition 27.
LetX be a BCC-algebra and let D be a generalized derivation on X. If x,y∈FixD(X), we have x∧y∈FixD(X).Proof.
Letx,y∈FixD(X), Then D(x)=x and D(y)=y.
From Proposition26, we have y*x∈FixD(X), and so d(y*x)=y*x. Hence, we have
(13)D(x∧y)=D(y*(y*x))D(x∧y)=(D(y)*(y*x))∧(y*d(y*x))D(x∧y)=(y*(y*x))∧(y*(y*x))D(x∧y)=y*(y*x)=x∧y.
Hence,x∧y∈FixD(X).



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