The Research On Pomonoid Actions And Fuzzy Rough Sets

Ever since fuzzy order was proposed by Zadeh in1971, various kinds of fuzzy orders have been introduced and have been widely applied in computer science. In order to investigate quantitative domain theory via fuzzy sets, Zhang and Fan introduced fuzzy partial order based on frame. It is equivalent to the L-order of Belohlavek. Fuzzy equivalence relations on fuzzy poset which are compatible with the fuzzy partial order are called Fuzzy partial order congruences. The correspon-dence between the fuzzy partial order congruences and the fuzzy order-preserving maps is discussed. We focus on the characterization of fuzzy partial order con-gruences of the fuzzy poset in terms of the fuzzy pseudo orders. The set of fuzzy pseudo orders of a fuzzy poset is a complete lattice. At last, fuzzy complete con-gruences of fuzzy complete lattice are discussed.Fuzzy partially ordered monoid is also explored in this study. We investigated the fuzzy partially ordered monoid and its representations in the categorical view. We proposed the fuzzy partially ordered monoid actions and S-fposet, where S is a fuzzy partially ordered monoid. In order to investigate the coproduct, pushout, we discuss the S-fposet congrunces. Then we investigate the adjoint situation of the category of S-fposet and the category of fuzzy posets.The essence of the monoid actions is a functor in the categorical view:from the category monoid to Set. Certainly, the functor that is from a small category to Set can be seen as a generalization of monoid actions. Motivated by Giry’s random topological actions, we propose the monoid random actions on dcpos and random actions of a small category on dcpos. The monoid random actions are based on the probabilistic power monad over the category D of dcpos. We investigate their properties and the adjoint situations between the category of random monoid actions RMrD and the base category Dε-Kleisli category of the probabilistic power monad (ε,η,τ). The random monoid actions could be seen as the Eilenberg-Moore category of a certain monad, which are algebras essentially. The congruence for the random monoid actions is also discussed. We put forward the concept of U-congruence on posets and discuss this kind of congruence for the monoid dcpos, random monoid dcpos, which are dcpos equipped with monoid actions and random monoid action respectively. Also the cartesian closedness of the category of random monoid actions over dcpos is investigated. At last, we generalize the random monoid actions to the functor case, i.e. it could be seen as a functor, and some properties of this category are investigated.Rough sets theory is established to deal with uncertain and imprecision in-formation, fuzzy sets theory is built to handle information that is without sharp boundaries. They are complemented to each other. With the development of rough set theory, classical rough set theory can not handle more complex problems, so various fuzzy generalizations of rough approximations have been proposed in the literature. We focus on discussion of the relationship between L-fuzzy rough sets and L-topologies on an arbitrary universe. Then one-to-one correspondence be-tween the set of all reflexive, transitive L-relations and the set of all Alexandrov L-topologies is obtained. Two kinds of homomorphisms of fuzzy approximation spaces based on complete residuated lattice are proposed. The homomorphisms are structure-preserving maps in some sense. We also introduce the fuzzy approx-imation subspaces and investigate their correspondence with the homomorphisms. Given a fuzzy equivalence relation, the factor set of the fuzzy approximation spaces are discussed. In recent years, rough sets on algebraic structures have developed quickly. At last, we introduce the rough sets on multilattices and investigate the properties of rough sub multilattices, rough ideals, rough filters. Generalized rough sets based on two multilattices are also discussed.This dissertation is supported by Hunan Provincial Innovation Foundation For Postgraduate.This dissertation is typeset by software LATEX2ε.