Attribute Reductions, Information-Systematic Homomorphisms And Rough Groups

Rough set theory, proposed by Z. Pawlak in the early 1980s, is a mathematicaltheory for reasoning about data. As a generalization of the classical set theory, thetheory of rough sets have successfully applied in many fields of artificial intelligence,such as machine learning, malfunction diagnosis, decision analysis, process controland knowledge discovery in databases. Therefore, the systematic and in-depth inves-tigation on rough sets has important value of theory and application. In this thesis,attribute reductions based on generalized rough sets are firstly studied. Then homo-morphisms between information systems and their properties are discussed. At last,some properties on the approximate structures of groups are investigated. The mainresults are summarized as follows:1. We introduce the notion of a relation decision system and study it's properties.By redefinitions of the classical concepts in generalized rough sets, the relationshipbetween the two forms of definition of approximations and some useful results fora relation decision system are derived. We study attribute reduction in a consistentrelation decision system, present a judgment theorem and a discernibility matrix as-sociated with attribute reduction and investigate the basic structure and properties ofthe attribute reduction.2. We investigate attribute reduction in a relation decision system from the gen-eral point of view, present a judgment theorem and a discernibility matrix associatedwith attribute reduction in a relation decision system and examine the basic struc-ture and properties of attribute reductions. The systematic approaches help us to gainmuch more insights into the mathematical structures of attribute reductions and pro-vide a theory basis for attribute reductions based on generalized rough set theory.3. We define two types of consistent functions and study their properties andintroduce the notions of relation mappings and discuss the problem of relation op-erations and set approximations under relation mappings. We propose notions ofhomomorphisms of information systems based on arbitrary binary relations, inves-tigate some properties of homomorphisms and prove that attribute reductions in theoriginal system and image system are equivalent to each other under the condition of homomorphism.4. We introduce the notions of consistent, strictly consistent and compatiblefunctions with respect to fuzzy relations, respectively and study their basic proper-ties. We define the concepts of fuzzy relation mappings and discuss the problem offuzzy relation operations and fuzzy set approximations under relation mappings. Wepropose the notions of homomorphisms of information systems based on fuzzy rela-tions, study their invariant properties and prove that attribute reductions in the fuzzyoriginal system and image system are equivalent to each other under the condition ofhomomorphism.5. We point out that some results on rough groups in some references are im-perfect and present the correct results and proofs. We then continue to discuss someproperties of approximate structures of rough groups, such as the relationships be-tween subgroups and their upper approximations and between normal subgroup se-quences and their upper approximations.