A NEW STUDY OF GENERALIZED  $\theta$--DERIVATION INVOLVING MAPPINGS  IN LATTICES

Aftab Ahmed Aftab; Mukhtar Ahmad; Muhammad Saleem; Muhammad Asif Javed; Muhammad Danial Faiz; Ather Qayyum

doi:10.5269/bspm.77829

Aftab Ahmed Aftab Department of Mathematics, University of Southern Punjab Multan, Pakistan
Mukhtar Ahmad Department of Mathematics, Khawaja Fareed University of Engineering and Information Technology Rahim Yar Khan, Pakistan
Muhammad Saleem National College of Business Administration and Economics Multan Campus, Pakistan
Muhammad Asif Javed Department of Mathematics, University of Southern Punjab Multan, Pakistan.
Muhammad Danial Faiz Department of Mathematics, University of Southern Punjab Multan, Pakistan.
Ather Qayyum Department of Mathematics, University of Southern Punjab Multan, Pakistan.

Résumé

The primary objective of this study is to explore and establish significant structural properties of lattices in the context of $\theta$-generalized derivations. We investigate the interplay between mappings and the intrinsic order structure of lattices, with particular emphasis on functions and their associated mappings. A central focus is the development and representation of $\theta$-generalized derivations involving left centralizers. Furthermore, we rigorously analyze and prove several fundamental results that elucidate the behavior and characteristics of these derivations within lattice frameworks. This work contributes to a deeper understanding of algebraic structures governed by generalized derivations and their functional mappings in ordered systems.

Téléchargements

Les données sur le téléchargement ne sont pas encore disponible.

Références

[1] Asci, M. and Ceran, S., (2013). Generalized (f g)-derivations of lattices. Math. Science and Applications E-Notes,2:56-62.
[2] Balbes, R. and Dwinger, P., (1974). Distributive Lattices. University of Missouri Press, Columbia, United State.
[3] Bell, A.J., (2003). The co-information lattice. in: 4th International Symposium on Independent Component Analysis and Blind Signal Sepration (ICA2003), Nara, Japan:921-926.
[4] Birkhoof, G., (1940). Lattice Theory.American Mathematical Society, New York.
[5] Carpineto. C and Romano. G, (1996). Information retrieval through hybird navigation of lattice representations. Int. J. Human Computers Studies, 45:553-578.
[6] Ceven, Y. and Öztürk, M.A., (2008). On f-derivations of lattices. Bull. Korean Math. Soc., 45:701-707.
[7] Chaudhary, M.A. and Ullah, Z.,(2011). On generalized ( )-derivations on lattices. Quaest. Math., 34:417-424.
[8] Degang, C., Wenxiu, Z., Yeung, D. and Tsang, E.C.C., (2006). Rough approximations on a complete distributive lattice with application to generalized rough sets. Informat. Sci., 176:1829-1848.
[9] Durfee, G., (2002). Cryptanalysis of RSA using algebraic and lattice methods. A dissertation submitted to the Department of Computer Sciences and the committe on graduate studies of Stanford University:1-114.
[10] Honda, A. and Grabisch, M., (2006). Entropy of capacities on lattices and set systems. Informat. Sci., 176:3472-3489.
[11] Posner, E., (1957). Derivations in prime rings. Proc. Amer. Math. Soc., 8:1093-1100.
[12] Sandhu, R.S., (1996). Role hierarchies and constraints for lattice-based access controls. in: Proceeding of the 4th European Symposium on Research in computer Security, Rome, Italy :65-79.
[13] Szasz, G., Derivations of lattices, (1975). Acta. Sci. Math.(Szeged) , 37:149-154.
[14] Ullah, Z., Javaid, I. and Chaudhary, M.A., (2013). Derivations involving centralizers. Util. Math., 90:383-392.
[15] Ullah, Z., Javaid, I. and Chaudhary, M.A., (2014). On generalized triple derivations on lattices. Ars Combin., 113:463-471.
[16] Xin, X.L., Liu, T.Y. and Lu, J.H., (2008). On derivations of lattices. Informat. Sci., 178:307-316.
[17] Xin, X.L., (2012). The xed set of a derivations in lattices. Fixed point theory and its application, 23:2012-218.
[18] Zalar, B., (1991). On centralizers of semiprime rings. Comment. Math. Univ. Carolinae, 4:609-614.
[19] Zhan, J. and Liu, T.Y., (2005). On f-derivations of BCI-algebras. Int. J. Math. Sci., 11:1675-1684