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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 238490, 6 pages
On Symmetric Left Bi-Derivations in BCI-Algebras
1Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
2Department of Mathematics Education (and RINS), Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics, Faculty of Science, Suleyman Demirel University, 32260 Isparta, Turkey
Received 15 February 2013; Accepted 30 May 2013
Academic Editor: Aloys Krieg
Copyright © 2013 G. Muhiuddin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The notion of symmetric 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.
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 ). Hence, most of the algebras related to the-norm-based logic such as MTL , BL, hoop, MV  (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 ). This shows that BCK-/BCI-algebras are considerably general structures.
Throughout our discussion,will denote a BCI-algebra unless otherwise mentioned. In the year 2004, Jun and Xin  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 a-semisimple BCI-algebra. For a self-mapof a BCI-algebra, they defined a-invariant ideal and gave conditions for an ideal to be-invariant. According to Jun and Xin, a self mapis called a left-right derivation (briefly-derivation) ofifholds for all. Similarly, a self mapis called a right-left derivation (briefly-derivation) ofifholds for all. Moreover, ifis both- and-derivation, it is a derivation on. After the work of Jun and Xin , 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 of-derivation , left derivation , 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 , Zhan and Liu have given the notion of-derivation of BCI-algebras as follows: a self mapis said to be a left-right-derivation or--derivation ofif it satisfies the identityfor all. Similarly, a self mapis said to be a right-left-derivation or--derivation ofif it satisfies the identityfor all. Moreover, ifis both- and--derivation, it is said thatis an-derivation, whereis an endomorphism. In the year 2007, Abujabal and Al-Shehri  defined and studied the notion of left derivation of BCI-algebras as follows: a self mapis said to be a left derivation ofif satisfyingfor all. Furthermore, in 2011 , Ilbira et al. have introduced the notion of symmetric bi-derivations in BCI-algebras. Following , a mappingis said to be symmetric ifholds for all pairsA symmetric mappingis called left-right symmetric bi-derivation (briefly-symmetric bi-derivation) if it satisfies the identityfor all.is called right-left symmetric bi-derivation (briefly-symmetric bi-derivation) if it satisfies the identityfor all. Moreover, ifis both a- and a-symmetric bi-derivation, it is said thatis a symmetric bi-derivation on.
Motivated by the notion of symmetric bi-derivations  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 and-regular symmetric left bi-derivations. Finally, we characterize the notion of-semisimple BCI-algebraby using the concept of symmetric left bi-derivation and show that, in a -semisimple BCI-algebrais a symmetric left bi-derivation if and only if it is a symmetric bi-derivation.
We begin with the following definitions and properties that will be needed in the sequel.
A nonempty setwith a constantand a binary operationis called a BCI-algebra if for allthe following conditions hold: (I)(II)(III)(IV)andimply
Define a binary relationonby lettingif and only if. Thenis a partially ordered set. A BCI-algebrasatisfyingfor all is called BCK-algebra.
A-algebrahas the following properties for all(a1). (a2)(a3)impliesand. (a4)(a5)(a6)(a7)implies
For a BCI-algebra, denote by(resp., ) the-part (resp., the-G part) of; that is,is the set of allsuch that(resp., ). Note that(see ). If, thenis called a-semisimple -algebra. In a-semisimple-algebra, the following hold. (a8). (a9)for all. (a10). (a11)implies. (a12)implies. (a13)implies. (a14). (a15).
Letbe a-semisimple BCI-algebra. We define addition “+” asfor all. Thenis an abelian group with identityand. Conversely, letbe an abelian group with identity, and let. Thenis a-semisimple BCI-algebra andfor all(see ).
For a BCI-algebra we denote, in particular, andWe call the elements ofthe-atoms of. For any, let, which is called the branch ofwith respect to. It follows thatwheneverandfor alland all. Note thatwhich is the-semisimple part of, andis a-semisimple BCI-algebra if and only if(see [23, Proposition 3.2]). Note also that; that is,, which implies thatfor all. It is clear that, andandfor alland all. Letbe a symmetric mapping. Then for all, a mappingdefined byis called trace of. For more details, refer to [3, 4, 6, 11, 22, 23].
3. Symmetric Left Bi-Derivations
The following definition introduces the notion of symmetric left bi-derivation for a BCI-algebra.
Definition 1. A symmetric mappingis called a symmetric left bi-derivation ofif it satisfies the following identity:
Example 2 (see ). Consider a-semisimple BCI-algebrawith the following Cayley table:
Define a mappingby
It is routine to verify thatis a symmetric left bi-derivation of
Theorem 3. Letbe a symmetric left bi-derivation ofThen (1)(2)(3)(4)
Proof. (1) LetThen, and so
(2) For anyimpliesand so
(3) By (2), we have. Then
(4) For anyand, we have This completes the proof.
Using Theorem 3, we have the following corollary.
Corollary 4. Letbe a symmetric left bi-derivation andbe the trace ofThen (1)(2)
Theorem 5. Letbe a symmetric left bi-derivation ofThen (1)(2)(3)(4)
Proof. (1) Let. Then
(2) Supposefor all. It is clear that, for, we haveConversely let us assume thatthen by using Theorem 3(4), we have.
(3) For any, we have
(4) For any , we have Thus, we can writefor anyThis completes the proof.
Definition 6. A symmetric left bi-derivation of a BCI-algebra is said to be componentwise regular if for all . In particular, is called -regular if .
Theorem 7. Letbe a symmetric left bi-derivation of BCI-algebra. Thenis a BCK-algebra if and only ifis componentwise regular symmetric left bi-derivation.
Proof. Supposeis a BCK-algebra. Then for any, we have
Henceis componentwise regular.
Conversely, letbe a componentwise regular symmetric left bi-derivation. Let for anybe such that. Then But it is clear that which is not possible asis a componentwise regular symmetric left bi-derivation. Thusis the unique-atom. Assume that for some, we have, then, so , which is a contradiction. Henceforth, for all, we haveimplying thereby,is a BCK-algebra.
This completes the proof.
Theorem 8. Letbe a componentwise regular symmetric left bi-derivation of a BCI-algebraThen (1)Bothandbelong to the same branch for all(2)(3)
Proof. (1) For any, we get
(2) Sinceis componentwise regular,Then
(3) Sincefor allby (2), using (a3) we obtain This completes the proof.
Next, we prove some results in a-semisimple BCI-algebra.
Theorem 9. Letbe a symmetric left bi-derivation of a-semisimple BCI-algebra; one has the following assertions.(1)(2)(3)
Proof. (1) Letbe a-semisimple BCI-algebra. Then for any, we have
(2) Let. Using (I), we have These above inequalities can be rewritten as Consequently, we get Also, using Theorem 5(4) and (1), we obtain
Sinceis a-semisimple BCI-algebra, hence, by using (21) and (a12), the above (20) yields.
(3) We haveby Theorem 5(4). Further, on letting, we get that implies. Henceforth, which amounts to say that.
This completes the proof.
Theorem 10. Letbe a-semisimple BCI-algebra. Thenis a symmetric left bi-derivation if and only if it is a symmetric bi-derivation on.
Proof. Suppose thatis a symmetric left bi-derivation onFirst, we show thatis a (r,l)-symmetric bi-derivation on. Let. Using Theorem 9(1) and (a14), we have
Henceis a-symmetric bi-derivation on.
Again, we show thatis a (l,r)-symmetric bi-derivation on. Let. Using Theorem 9(1), (3) and (a15), we have
Conversely, suppose thatis a symmetric bi-derivation ofAsis a (r,l)-symmetric bi-derivation on, then for anyand using (a14), we have Henceis a symmetric left bi-derivation. This completes the proof.
The authors are grateful to the anonymous referee(s) for a careful checking of the details and for helpful comments that improved the present paper. G. Muhiuddin and Abdullah M. Al-roqi were partially supported by the Deanship of Scientific Research, University of Tabuk, Ministry of Higher Education, Saudi Arabia.
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