Algorithm and axiomatization of rough fuzzy sets
based finite dimensional fuzzy vectors |
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| Authors: | Mingfen WU |
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| Affiliation: | School of Information
Science, Wuyi University, Jiangmen 529020, China；Key Laboratory
of Intelligent Information Processing, Institute of Computing Technology,
Chinese Academy of Sciences, Beijing 100190, China; |
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| Abstract: | Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system. |
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| Keywords: | fuzzy vector fuzzy matrix fuzzy relation operation basis lower and upper approximation rough fuzzy sets unary operator |
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