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This article deals with approaches to knowledge reductions in inconsistent information systems (ISs). The main objective of this work was to introduce a new kind of knowledge reduction called a maximum distribution reduct, which preserves all maximum decision classes. This type of reduction eliminates the harsh requirements of the distribution reduct and overcomes the drawback of the possible reduct that the derived decision rules may be incompatible with the ones derived from the original system. Then, the relationships among the maximum distribution reduct, the distribution reduct, and the possible reduct were discussed. The judgement theorems and discernibility matrices associated with the three reductions were examined, from which we can obtain approaches to knowledge reductions in rough set theory (RST).

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Zadeh's seminal work in theory of fuzzy-information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it. This has led to an interesting research topic: granular computing (GrC). Although many excellent research contributions have been made, there remains an important issue to be addressed: What is the essence of measuring a fuzzy-information granularity of a fuzzygranular structure? What is needed to answer this question is an axiomatic constraint with a partial-order relation that is defined in terms of the size of each fuzzy-information granule from a fuzzy-binary granular structure. This viewpoint is demonstrated for fuzzy-binary granular structure, which is called the binary GrC model by Lin. We study this viewpoint from from five aspects in this study, which are fuzzy BINARY-granular-structure operators, partial-order relations, measures for fuzzy-information granularity, an axiomatic approach to fuzzy-information granularity, and fuzzy-information entropies.

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