| The rough set theory is a branch of Mathematic to study uncertain knowledge. It can analysis and process effectively all kinds of immaturity information as inexactness, uncertain, uncompleted, find implicative knowledge and open out potential rules. it is widely applied in many fields, such as information, engineering and technology. So science appeared in 80's last century, it has been developed quickly. Recently, the rough set theory was combined with algebra structures, topology structures and order structures, and then many articles in these fields were appeared. Kuroki N firstly put forward the notion of rough subsets and rough ideals in semigroups, testifying that Under the condition of the congruence relation, a rough set of a subsemigroup was proved to be its subsemigroup, while that of a left (right, bisides)ideal was also proved to be its left (right, bisides)ideal. Next, rough sets in a group have been studied and the concepts of Rough subgroups, Rough normal subgroups been first introduced. Consequently, Kuroki N studied some essential properties about them.Fuzzy mathematics is also a new branch of Mathematics to study uncertain problems(Fuzzy uncertain problems). For the needs in engineering fields, Zhde, a famous American electron engineer and cybernetics expert, put forward the theory of Fuzzy set. This theory not only expends the mathematics bases, but also bring many mathematics methods and theories.Today ,Fuzzy set theories, rough set theory was combined with algebra structures, which gave birth to Fuzzy rough algebra. Zhang and Xiao popularized rough structures from semigroups to Fuzzy semigroups, showing the definition of Fuzzy rough semigroup, Fuzzy rough sub ring and Fuzzy rough ideals by congruence, and then studied their properties. In the following, by Fuzzy congruence, Zhao defined Fuzzy rough sub set in Fuzzy sets and studied their properties.In this thesis, we defined duality Fuzzy relation in a pointwise Fuzzy set by a set of duality Fuzzy point. Then we defined Fuzzy congruence and Fuzzy rough set in Fuzzy semigroups.There have six chapters in this paper. The first chapter is Introduction, the theory of Fuzzy set and rough set are introduced simply. The second chapter is introduction, showing the method of establishing pointwise Fuzzy set. The third chapter is Fuzzy congruence in... |
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