Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 238490, 6 pages
http://dx.doi.org/10.1155/2013/238490
Research Article
On Symmetric Left Bi-Derivations in BCI -Algebras
G. Muhiuddin,
1
Abdullah M. Al-roqi,
1
Young Bae Jun,
2
and Yilmaz Ceven
3
1
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
2
Department of Mathematics Education (and RINS), Gyeongsang National University, Jinju 660-701, Republic of Korea
3
Department of Mathematics, Faculty of Science, Suleyman Demirel University, 32260 Isparta, Turkey
Correspondence should be addressed to G. Muhiuddin; chishtygm@gmail.com
Received 15 February 2013; Accepted 30 May 2013
Academic Editor: Aloys Krieg
Copyright © 2013 G. Muhiuddin et al. Tis 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.
Te notion of symmetric lef bi-derivation of a BCI -algebra X is introduced, and related properties are investigated. Some results
on componentwise regular and d-regular symmetric lef 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 lef bi-derivation if and only if it is
a symmetric bi-derivation.
1. Introduction
BCK-algebras and BCI -algebras are two classes of nonclassi-
cal logic algebras which were introduced by Imai and Is´ eki
in 1966 [1, 2]. Tey 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). Te 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 the -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]). Tis shows that BCK-/BCI -algebras are
considerably general structures.
Troughout our discussion, will 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 -algebras, and as a result they introduced a
new concept, called a (regular) derivation, in -algebras.
Using this concept as defned they investigated some of its
properties. Using the notion of a regular derivation, they
also established characterizations of a -semisimple -
algebra. For a self-map of a -algebra, they defned a -
invariant ideal and gave conditions for an ideal to be -
invariant. According to Jun and Xin, a self map :→
is called a lef-right derivation (briefy (,)-derivation)
of if (∗)=()∗∧∗() holds for all ,∈.
Similarly, a self map :→ is called a right-lef
derivation (briefy (,)-derivation) of if ( ∗ ) = ∗
() ∧ () ∗ holds for all , ∈ . Moreover, if is
both (,)- and (,)-derivation, it is a derivation on . Afer
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 of -derivation [18], lef deriva-
tion [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 of -derivation of BCI -algebras as follows: a self
map
:→ is said to be a lef-right -derivation or
(,)--derivation of if it satisfes the identity
( ∗
) =
() ∗ () ∧ () ∗
() for all , ∈ .
Similarly, a self map
:→ is said to be a right-
lef -derivation or (,)--derivation of if it satisfes the
identity
( ∗ ) = () ∗
() ∧
() ∗ () for all
,∈. Moreover, if
is both (,)- and (,)--deriva-
tion, it is said that
is an -derivation, where is an
endomorphism. In the year 2007, Abujabal and Al-Shehri
[12] defned and studied the notion of lef derivation of BCI -
algebras as follows: a self map :→ is said to be a lef