GENERALIZED (f; g)-DERIVATIONS OF LATTICES

Year 2013, Volume: 1 Issue: 2, 56 - 62, 01.12.2013

Mustafa Aşcı Şahinceran

Abstract

In this paper as a generalization of derivation and f -derivation ona lattice we introduce the notion of generalized (f, g)-derivation of a lattice.We give illustrative example. If the function g is equal to the function f thenthe generalized (f, g)-derivation is the f -derivation defined in [8]. Also if wechoose the function f and g the identity functions both then the derivation wedefine coincides with the derivation defined in [22]

Keywords

Lattice , Derivation , (f , g)-derivation

References

• A. J. Bell, The co-information lattice, 4th Int. Symposium on Independent ComponentvAnal- ysis and Blind Signal Seperation (ICA2003), Nara, Japan, 2003, 921–926.
• A. Honda and M. Grabisch, Entropy of capacities on lattices and set systems, Inform. Sci. 176 (2006), no. 23, 3472–3489.
• Asci, M., Kecilioglu O., Davvaz B. On Symmetric f bi-derivations of Lattices. J. Combin. Math. Combin. Comput. 83, (2012), pp. 243-253..
• Asci, M., Kecilioglu O., Ceran S¸. Permuting Tri (f,g) derivations on Lattices. Ann. Fuzzy Math. Inform. (AFMI). Vol 1, No.2 (2011), pp. 189-196. .
• Ceran, S¸. Asci, M. Symmetric bi-(σ, τ ) derivations of prime and semi prime gamma rings. Bull. Korean Math. Soc. 43 (2006), no. 1, 9–16.
• C. Carpineto and G. Romano, Information retrieval through hybrid navigation of lattice representations, International Journal of Human-Computers Studies 45 (1996), 553–578.
• C. Degang, Z. Wenxiu, D. Yeung, and E. C. C. Tsang, Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Inform. Sci. 176 (2006), no. 13, 1829–1848.
• C¸ even, Y. ¨Ozt¨urk, M. A. On f-derivations of lattices. Bull. Korean Math. Soc. 45 (2008), no. 4, 701–707.
• C¸ even, Y. Symmetric bi derivations of Lattices, Quaestiones Mathematicae, 32(2009), 1-5
• C¸ even, Y. ¨Ozt¨urk, M. A. Some properties of symmetric bi-(σ, τ )-derivations in near-rings. Commun. Korean Math. Soc. 22 (2007), no. 4, 487–491.
• Davey, B. A.; Priestley, H. A. Introduction to lattices and order. Second edition. Cambridge University Press, New York, 2002. xii+298 pp. ISBN: 0-521-78451-4
• E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.
• Ferrari, Luca On derivations of lattices. Pure Math. Appl. 12 (2001), no. 4, 365–382.
• G. Durfee, Cryptanalysis of RSA using algebraic and lattice methods, A dissertation submit- ted to the department of computer sciences and the committe on graduate studies of Stanford University (2002), 1–114.
• G. Birkhoof, Lattice Theory, American Mathematical Society, New York, 1940.
• H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (1989), no. 3-4, 339–346.
• J. Zhan and Y. L. Liu, On f-derivations of BCI-algebras, Int. J. Math. Math. Sci. (2005), no. 11, 1675–1684.
• Ozbal, S.A, Firat, A. Symmetric f bi Derivations of Lattices. Ars Combin. 97 (2010) in press. [19] R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, Mo., 1974.
• R. S. Sandhu, Role hierarchies and constraints for lattice-based access controls, Proceedings of the 4th Europan Symposium on Research in Computer Security, Rome, Italy, 1996, 65–79. [21] Sz´asz, G. Derivations of lattices. Acta Sci. Math. (Szeged) 37 (1975), 149–154.
• X. L. Xin, T. Y. Li, and J. H. Lu, On derivations of lattices, Inform. Sci. 178 (2008), no. 2, 307–316. [23] Y. B. Jun and X. L. Xin, On derivations of BCI-algebras, Inform. Sci. 159 (2004), no. 3-4, 167–176. Pamukkale University Science and Arts Faculty Department of Mathematics Kınıklı Denizli TURKEY
• E-mail address: [e-mail]
• E-mail address: [e-mail]