The Research Of Rough Sets Generated By Ideal On Lattices And Lattice-valued Rough Sets

Generalization of rough set models is an important aspect of studying rough set theory. Using algebraic systems to extend rough set theory becomes significant important in this research field. In 2006, Chen et al. selected the complete completely distributive lattice(CCD lattice for short) as the basic algebra system for defining more general and abstract approximation operators, which generated from the notion of cover introduced on a CCD lattice. In 2013, Qin et al. proposed a kind of lower approximation operator and three kinds of upper approximation operators on a CCD lattice based on the neighborhood with respect to a cover, and the relationship among the approximation operators were investigated. In another way, Zhou and Hu employed binary relations on CCD lattices to construct the lower and upper approximation operators in 2014.In this paper, we define new approximation operators in terms of an ideal on a CCD lattice based on a binary relation and the neighborhood with respect to a cover, respectively.In the second chapter, based on a binary relation R on a CCD lattice L, a pair of approximation operators via an ideal I of L are introduced, which can be viewed as a generalization of Zhou and Hu’s approximation operators. When I is the least ideal of L and R a reflexive binary relation on L, these two approximations coincide. Let L be a complete atomic Boolean lattice and R a reflexive and transitive binary relation on L, the characterizations in terms of the ideal are presented when these two approximations coincide. New rough approximations make the accuracy measures are higher than the existing approximations, and this result in terms of the ideal is also interpreted by an example. Moreover, the topological and lattice structures of the approximation operators are given.In the third chapter, based on the neighborhood with respect to a cover on a CCD lattice L, we define the lower and upper approximation operators using an ideal of L. Which can be seen as an extension of the approximation operators introduced by Qin et al. Our rough approximations improve the existing covering approximations on a CCD lattice. When L is a power set lattice, the approximations based on the adhesion are special cases of our approximations. The relationships between new approximations and the old covering approximations are discussed.The fourth chapter gives a further study on φ-fuzzy rough sets. The wellknown R-fuzzy rough sets can be treated as a special φ-fuzzy rough sets. First, by means of the idea of level sets, we define a family of L-fuzzy relations and thereby give the representation of φ-fuzzy rough sets, that is, φ-fuzzy rough sets can be represented by a family of R-fuzzy rough sets. Second, we obtain an intuitive interpretation of the function φ, which can be regarded as a generalization of the ordinary neighborhood operator in L-fuzzy settings. Then we discuss the corresponding φ-fuzzy rough sets by defining the special function φ. Lastly, rather than restricting the lattice-context, we study the induced L-topology by restricting the function φ.The fifth chapter investigates the lattice-value ideals of a Quantale. Considering L be a complete residuated lattice, we propose several characterizations of L-fuzzy ideals of a Quantale in terms of fuzzy points and level sets. In particular,we make use of R-fuzzy rough sets induced by L-fuzzy sets to characterize L-fuzzy ideals of a Quantale. Moreover, we introduce the concept of L-fuzzy quasi-rough ideals of a Quantale, and discuss the connections among L-fuzzy rough ideals,L-fuzzy quasi-rough ideals and L-fuzzy ideals.