Methods for knowledge reduction in inconsistent ordered information systems

Knowledge reduction is one of the most important problems in the study of rough set theory. However, in real-world, most of information systems are based on dominance relations in stead of the classical equivalence relation because of various factors. The ordering of properties of attributes plays a crucial role in those systems. To acquire brief decision rules from the systems, knowledge reductions are needed. The main objective of this paper is to deal with this problem. The distribution reduction and maximum distribution reduction are proposed in inconsistent ordered information systems. Moreover, properties and relationship between them are discussed. Furthermore, judgment theorem and discernibility matrix are obtained, from which an approach to knowledge reductions can be provided in inconsistent ordered information systems.