SYMMETRIC BI-T-DERIVATION OF LATTICES

Year 2019, Volume: 9 Issue: 3, 554 - 562, 01.09.2019

C. Jana K. Hayat M. Pal

Abstract

In this paper, the notion of a new kind of derivation is introduced for a lattice L;_;^ , called symmetric bi-T-derivations on L as a generalization of derivation of lattices and characterized some of its related properties. Some equivalent conditions provided for a lattice L with greatest element 1 by the notion of isotone symmetric bi-T-derivation on L. By using the concept of isotone derivation, we characterized the modular and distributive lattices by the notion of isotone symmetric bi-T-derivation on L.

Keywords

Lattice, Derivation of lattice, Symmetric bi-T-derivation of lattice, Modular lattice and Distributive lattice.

References

• Birkhoof, G., (1940), Lattice Theory, Amer. Math. Soc., New York.
• Balbes, R. and Dwinger, P., (1974), Distributive Lattices, University of Missouri Press, Columbia, USA.
• Bell, A. J., (2003), The co-information lattice, in: 4th International Sympo- sium on Independent Com- ponent Analysis and Blind Signal Separation (ICA2003), Nara, Japan, pp. 921-926.
• Carpineto, C. and Romano, G., (1996), Information retrieval through hybrid naviga- tion of lattice representations, Int. J. Human Computers Studies, 45, pp. 553-578.
• Sandhu, R.S., (1996), Role hierarchies and constraints for lattice-based access con- trols, in: Proceedings of the 4th European Symposium on Research in Computer Security, Rome, Italy, pp. 65-79.
• Durfee, G., (2002), Cryptanalysis of RSA using algebraic and lattice methods, A dissertation submitted to the Department of Computer Science and the committee on graduate studies of Stanford University, pp. 1-114.
• Honda, A. and Grabisch, M., (2006), Entropy of capacities on lattices and set systems, Inform. Sci., 176, pp. 3472-3489.
• Posner, E., (1957), Derivations in prime rings, Proc. Am. Math. Soc., 8, pp. 1093-1100.
• Bell, H. E. and Kappe, L. C., (1989), Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53 (3-4), pp. 339-346.
• Bresar, M., (1991), On the distance of the composition of the two derivations to the generalized deriva- tions, Glasgow Math. J., 33 (1), pp. 89-93.
• Hvala, B., (1998), Generalized derivations in rings, Common. Alg., 26 (4), pp. 1147-1166.
• Arga¸c, N. and Albas, E., (2004), Generalized derivations of prime rings, Algebra Coll., 11, pp. 399-410.
• G¨olba¸si, ¨O. and Kaya, K., (2006), On Lie ideal with generalized derivations, Siberian. Math. J., 47 (5), pp. 862-866.
• Jana, C., Senapati, T. and Pal, M., (2016), (∈, ∈ ∨q)-intuitionistic fuzzy BCI-subalgebras of BCI- algebra, Journal of Intelligent and Fuzzy systems, 31, pp. 613-621.
• Jana, C., Senapati, T., Bhowmik, M. and Pal, M., (2015), On intuitionistic fuzzy G-subalgebras ofG-algebras, Fuzzy Information and Engineering, 7, pp. 195-209.
• Jana, C. and Pal, M., (2016), Applications of new soft intersection set on groups, Annals of Fuzzy Mathematics and Informatics, 11 (6), pp. 923-944.
• Jana, C., (2015), Generalized (Γ, Υ)-derivation on subtraction algebras, Journal of Mathematics and Informatics, 4, pp. 71-80.
• Jana, C. and Pal, M., (2017), Application of (α, β)-soft intersectional sets on BCK/BCI-algebras, Int. J. Intelligent Systems Technologies and Applications, 16 (3), pp. 269-288.
• Jana, C., Pal, M. and Saied, A. B., (2017), (∈, ∈ ∨q)-bipolar fuzzy BCK/BCI-algebras, Missouri Journal of Mathematical Scienc, (accepted).
• Jana, C., Senapati, T. and Pal, M., (2015), Derivation, f -derivation and generalized derivation of KU S- algebras, Cogent Mathematics, 2, pp. 1-12.
• Jana, C., Senapati, T. amd Pal, M. , (2017), On t-derivation of complicated subtraction algebras, Journal of Discrete Mathematical Sciences and Cryptography, (accepted).
• Xin, X. L., Li, T. Y. and Lu, J. H., (2008), On Derivations of Lattices, Inform. Sci., 178, pp. 307-316.
• Maksa, G. Y., (1980), A remark on symmetric biadditive functions having nonnegative diagonalization, Glasnik Math, 15 (35), pp. 279-282.
• Maksa, G. Y., (1989), On the trace of symmetric bi-derivations, C.R. Math. Rep. Acad. Sci. Canada, 9, pp. 303-307.
• Ozturk, M.A. and Sapancy, M., (1999), On generalized symmetric bi-derivations in prime rings, East
• Asian Mathematical Journal, 15 (2), pp. 165-176. Sapancy, M., Ozturk, M. A. and Jun, Y. B., (1999), Symmetric bi-derivations on prime rings, East Asian
• Mathematical Journal, 15 (1), pp. 105-109. Vukman, J., (1989), Symmetric bi-derivations on prime and semi-prime rings, Aequationes Mathemati- cae, 38, pp. 245-254.
• Vukman, J., (1990), Two results concerning symmetric bi-derivations on prime rings, Aequationes Math- ematicae, 40, pp. 181-189.
• C¸ ven, Y., (2009), Symmetric bi-derivations of lattices, Quaest. Math., 32, pp. 241-245.