Characterization of Associative PU-algebras by the Notion of Derivations

: In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.


Introduction
In 1966, Y. Imai and K. Isèki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras [1][2][3]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. Neggers et al. [4] introduced a notions, called Q-algebras, which is a generalization of BCH / BCI / BCK-algebras and generalized some theorems discussed in BCIalgebras. Megalai and Tamilarasi [5] laid down the foundation of a notion, called TM-algebra. Moreover, Mostafa et al. [6] introduced a new algebraic structure called PU-algebra, which is a dual for TM-algebra and investigated severed basic properties. Moreover they derived new view of several ideals on PUalgebra and studied some properties of them. Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. Several authors [7][8][9][10][11] have studied derivations in rings and near rings. Jun and Xin [12] applied the notions of derivations in ring and near-ring theory to BCI-algebras, and they also introduced a new concept called a regular Derivation in BCI-algebra. They investigated some of its properties, defined a d -derivations ideal and gave conditions for an ideal to be d-derivations. Later, Abujabal and Al-Shehri [13], defined a left derivations in BCI-algebras and investigated a regular left derivations. Zhan and Liu [14] studied f-derivations in BCI-algebras and proved some results. Muhiuddin and Alroqi [15,16] introduced the notions of (α, β)-derivations in a BCI-algebras and investigated related properties. They provided a condition for a (α, β) -derivations to be regular. They also introduced the concepts of a d (α, β) -invariant (α, β)derivations and α-ideal, and then they investigated their relations. Furthermore, they obtained some results on regular (α, β)derivations. Moreover, they studied the notions of t-derivations on BCI-algebras [17] and obtain some of its related properties. Further, they characterized the notions of p-semisimple BCIalgebras X by using the notions of t-derivations. Abujabal and Shehri in their pioneer paper [18], defined the derivations as, for a self-map, d, for any algebra X, d is a left-right derivation (briefly (l, r)-derivation) of X if it satisfies the identity d( . For all , X. If d satisfies the identity ( ) = ( ( )) ˄ ( ( ) ) for all , X, then d is a right-left derivation (briefly (r, l)-derivation) of X. If d is both (l, r)-derivation and (r, l)-derivation, then d is a derivation of X. The aim of the paper is to complete the studies on PU-algebra; in particular, we aim to apply the notion of derivation on associative PU-algebra and obtain some related properties. We start with definitions and propositions on PUalgebra taken from [6]. Then, we redefine the notion of derivation in associative PU-algebra and prove that for ф being a (Left, Right) or (Right, Left) -derivation of an associative PU-algebra Ⱬ then ф is one-one map. If ф is a regular map then it is identity. If there exists an element ∈ Ⱬ such that ф( )= then the map ф is identity. We prove that If ф is (Left, Right) -regular We prove that if ф is a self-map of an associative PU-Algebra Ⱬ then ( * ( * ф( ))) * =(ф( ) * (ф( ) * )) * . We also prove that if ф is a regular (Right, Left)derivation of an associative PU-algebra Ⱬ then Ker(ф)={ ∈ Ⱬ: ф( )=0} is a subalgebra of Ⱬ.

Preliminaries
This section consists of some preliminary definitions and basic facts about PU-algebra which are useful in the proofs of our results. Throughout this research work we denote the PUalgebra always by Ⱬ.

Main Results
Definition 3.1:-Let (Ⱬ, * , 0) is an associative PU-algebra From (P 9 ) we have Also from (P 9 ) we have From (2) and (3) we get Using the result of equation (1) in equation (4) we get By (P 8 ) left cancellation law holds in Ⱬ therefore from (5) we get = . Hence ф is one to one. Proof (P 11 ): Let ф is regular then we have From (P 9 ) we have From (6) and (7) we get Now by using (P 3 ) in the right hand side of equation (8) then (8) becomes By (P 8 ) right cancellation law holds in Ⱬ therefore (9) becomes ф( )= ∀ ∈ Ⱬ.

Conclusion
We see that derivations with special properties play a central role in the investigation of the structure of an algebraic system.
The forthcoming study of derivations in PU-algebras may be the following topics are worth to be taken into account.
To describe left derivations in PU-algebras and investigate a regular left derivations by using this concept.
To refer this concept to some other algebraic structures.
To consider the results of this concept to some possible applications in information systems and computer sciences.