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Abstract
Generalization of rough set model is an important aspect of rough set theory research. The problem to be discussed in this thesis is to minimize the boundary region and this requires a new approximation approach which increases lower approximation and decreases upper approximation. This will decrease the uncertainty region that help decision maker to get more accurate results. First we generalized the element based definition of the constructive method for defining rough sets by using two general relations. We defined two new approximations called bi lower and upper approximations and studied their properties. We generalized both element based definition and algebraic method for defining the theory of rough sets. Instead of one operation used by Jarvinen[24], we used two operations to define, in a lattice theoretical setting, two new mappings which mimic the rough approximations and called pairwise lower and pairwise upper approximations. We studied the properties of these approximations by imposing different axioms on the suggested two operations. Also, properties of the ordered set of the pairwise lower and upper of an element of a complete atomic Boolean lattice were investigated. Finally, we generalized the subsystem based definition of rough sets by using a topological structure generated from a general relation. We initiated two new operators based on the concepts of semi closure and semi interior [10] called semi lower and upper approximations. The number of possible membership relations was enlarged to four choices instead of two in [39] and [48, 49]. Many properties of the suggested operators were obtained. We introduced some new types of rough equality, rough inclusion, rough definability, rough undefinability, rough measures and rough edges based on these approximations.