ABSTRACT

In the year of 1982 Z Pawlak introduced idea of rough set. M. K. Sen introduced f -semigroups in 1981. The roughness of ideals in semi-groups was defined by N. Kuroki in 1997; consequently, the rough sub [-semi-group, rough ideal (R.I) and rough bi-ideal (R. bi-I) of f -semi-group were studied by Y. B. Jun in 2003. Q. Mei. Xiao and Zhen-Liang Zhang discussed the association between upper and lower rough prime ideals (upper and lower R.P ideal) of semi-groups. He also gave upper approximation and lower approximations (U-Approx and L-Approx, respectively) of their homomorphic images in a paper of 2006.
R. Chinram discussed the idea of rough prime ideals (R. P. Ideals) in f -semi-groups in 2009. Muhammad Shabir and Naila Kanwal introduced prime bi--ideals in semi-groups in 2007. Following those, Muhammad Shabir and Shujaat Ali Shah introduced prime bi-ideals of f -semi-groups in 2012. In this dissertation, we give the notion of rough prime bi-ideals (R.P bi-ideal) and semiprime bi-ideals of f -semi-groups and give some results about these R.P bi-ideals of f -semi-groups. Also, we incorporate upper and lower R. P ideal of f -semi-groups. Also, the U-Approx and L-Approx of their homomorphic images are studied. Moreover, we considered the upper and lower R.P bi-ideals of f -semi-groups and the U-Approx and L- Approx of their homomorphic.