Research On Generalized Approximation Space And Formal Context By Means Of Order And Topology

Rough set theory and formal concept analysis are mathematical tools applied to data analysis and knowledge processing.They are widely applied to machine learning,data mining and information processing now.Rough set theory and formal concept analysis are based on relation and partial order.So there are close relationships between them.Generalized approximation spaces and formal contexts are the core concepts of rough set theory and formal concept analysis respectively.In this paper,topology and Domain theory are applied to the research of generalized approximation spaces and formal contexts.These research work seeks a broader application field for topology and Domain theory,which is conducive to promoting the cross penetration between disciplines.This paper generalizes abstract bases to consistent F-augmented generalized approximation space(CF-approximation spaces,in short)and the family of round ideals of abstract base to the family of CF-closed sets of CF-approximation space in the framework of rough set theory.The relationships between CF-approximation spaces and continuous domains are studied.It is proved that CF-closed sets are special closed sets of the topology induced by relation.The properties about order structure of the family of CF closed sets are discussed.And then representation theorem of continuous domains is given via CF-closed sets.CF-approximation relations between CF-approximation spaces are defined,and categorical equivalence between categories of CF-approximation spaces with CF-approximation relations and categories of continuous domains with Scott continuous maps is established.This paper introduces relative reductions and topological reductions of knowledge bases which are generalizations of generalized approximation spaces.It is proved that topological reductions are special relative reductions relative to some preorder.The criterion theorem for relative reductions of finite knowledge bases is obtained by RMdiscernibility matrices.The abstract knowledge base(AKB,in short),which is a generalization of some covering generalized approximation space,is regarded as a poset endowed with set-inclusion order and then the relationship between the order structure characteristics of AKBs and the existence of finite join reductions is discussed.This paper generalizes join saturation reductions to relative join saturation reductions.The proof of the criterion theorem of relative join saturation reductions of finite AKBs is given by the properties of some Galois adjunction on AKBs.This paper introuduce the notion of AF-augmented formal context(AF-context,in short)and AF-concept set in the framework of formal concept analysis.This paper discusses the relationships between AF-contexts and arithmetic semilattices.The properties about order structure of the family of AF-concept sets are discussed.And then the representation theorem of arithmetic semilattices is given via AF-concept sets.AF-approximation relations between AF-contexts are defined,and categorical equivalence between categories of AF-contexts with AF-approximation relations and categories of arithmetic semilattices with Scott continuous maps is established.This paper defines AE-paracompactness of formal contexts by the idea of topology.It is proved that AE-paracompactness is preserved by certain information morphisms and that AEparacompactness is hereditary for certain closed embedded sub-contexts.It is proved that the object of a product of two AE-paracompact contexts is still AE-paracompact in category FCC.