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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

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

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.

1. Y. Imai and K. Iséki, “On axiom systems of propositional calculi. XIV,” Proceedings of the Japan Academy, vol. 42, pp. 19–22, 1966.
2. K. Iséki, “An algebra related with a propositional calculus,” Proceedings of the Japan Academy, vol. 42, pp. 26–29, 1966.
3. M. Aslam and A. B. Thaheem, “A note on $p$-semisimple BCI-algebras,” Mathematica Japonica, vol. 36, no. 1, pp. 39–45, 1991.
4. S. A. Bhatti, M. A. Chaudhry, and B. Ahmad, “On classification of BCI-algebras,” Mathematica Japonica, vol. 34, no. 6, pp. 865–876, 1989.
5. Q. P. Hu and K. Iséki, “On BCI-algebras satisfying $\left(x*y\right)*z=x*\left(y*z\right)$,” Kobe University. Mathematics Seminar Notes, vol. 8, no. 3, pp. 553–555, 1980.
6. M. Daoji, “BCI-algebras and abelian groups,” Mathematica Japonica, vol. 32, no. 5, pp. 693–696, 1987.
7. Q. P. Hu and K. Iséki, “On some classes of BCI-algebras,” Mathematica Japonica, vol. 29, no. 2, pp. 251–253, 1984.
8. F. Esteva and L. Godo, “Monoidal t-norm based logic: towards a logic for left-continuous t-norms,” Fuzzy Sets and Systems, vol. 124, no. 3, pp. 271–288, 2001.
9. C. C. Chang, “Algebraic analysis of many valued logics,” Transactions of the American Mathematical Society, vol. 88, pp. 467–490, 1958.
10. J. E. Whitesitt, Boolean Algebra and Its Applications, Dover, New York, NY, USA, 1995.
11. Y. B. Jun and X. L. Xin, “On derivations of BCI-algebras,” Information Sciences, vol. 159, no. 3-4, pp. 167–176, 2004.
12. H. A. S. Abujabal and N. O. Al-Shehri, “On left derivations of BCI-algebras,” Soochow Journal of Mathematics, vol. 33, no. 3, pp. 435–444, 2007.
13. S. Ilbira, A. Firat, and Y. B. Jun, “On symmetric bi-derivations of BCI-algebras,” Applied Mathematical Sciences, vol. 5, no. 57-60, pp. 2957–2966, 2011.
14. G. Muhiuddin and A. M. Al-roqi, “On $t$-derivations of BCI-algebras,” Abstract and Applied Analysis, vol. 2012, Article ID 872784, 12 pages, 2012.
15. G. Mudiuddin and A. M. Al-roqi, “On (α,β)-derivations in BCI-algebras,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 403209, 11 pages, 2012.
16. M. A. Öztürk, Y. Çeven, and Y. B. Jun, “Generalized derivations of BCI-algebras,” Honam Mathematical Journal, vol. 31, no. 4, pp. 601–609, 2009.
17. J. Zhan and Y. L. Liu, “On $f$-derivations of BCI-algebras,” International Journal of Mathematics and Mathematical Sciences, no. 11, pp. 1675–1684, 2005.
18. A. A. M. Kamal, “$\sigma$-derivations on prime near-rings,” Tamkang Journal of Mathematics, vol. 32, no. 2, pp. 89–93, 2001.
19. M. Brešar and J. Vukman, “On left derivations and related mappings,” Proceedings of the American Mathematical Society, vol. 110, no. 1, pp. 7–16, 1990.
20. J. Vukman, “Symmetric bi-derivations on prime and semi-prime rings,” Aequationes Mathematicae, vol. 38, no. 2-3, pp. 245–254, 1989.
21. J. Vukman, “Two results concerning symmetric bi-derivations on prime rings,” Aequationes Mathematicae, vol. 40, no. 2-3, pp. 181–189, 1990.
22. Y. B. Jun and E. H. Roh, “On the BCI-$G$ part of BCI-algebras,” Mathematica Japonica, vol. 38, no. 4, pp. 697–702, 1993.
23. Y. B. Jun, X. L. Xin, and E. H. Roh, “The role of atoms in BCI-algebras,” Soochow Journal of Mathematics, vol. 30, no. 4, pp. 491–506, 2004.
24. J. Meng, Y. B. Jun, and E. H. Roh, “BCI-algebras of order 6,” Mathematica Japonica, vol. 47, no. 1, pp. 33–43, 1998.