| Pawlak classic rough set is based on equivalence relation, then Dubois and Prade replaced theequivalence relation by fuzzy relation and proposed the notion of fuzzy rough set, then all kinds ofgeneralized rough set models were proposed and the axiomatic systems of these generalized roughset models were also introduced.Based on the research achieved at home and abroad, this paper proposed some newgeneralized rough set models, studies their basic properties and gives the axiomatic systems ofthem. The main results of this paper are as follows:(1) Study join and union closed problem of the fuzzy rough upper and lower approximationoperators, the rough implication operator based on fuzzy rough sets and its properties are given.With the help of the results and the approaches of generalized fuzzy rough sets on t-norm(triangular norm), this paper presents the axiomatic systems of fuzzy rough upper and lowerapproximation operators on T-similar relation.(2) Generalize the BCK-lattice, double BCK-lattice and involutive strong double BCK-latticeand their properties are given. The generalized rough set model based on involutive strong doubleBCK-lattice is established, and the basic properties are given. Discuss the rough implicationoperator based on generalized rough set model based on involutive strong double BCK-lattice andits properties. The axiomatic systems of fuzzy rough upper and lower approximation operatorsbased on involutive strong double BCK-lattice are presented.(3) The notion of regular commutative double residuated lattice is introduced, and thegeneralized rough set model based on regular commutative double residuated lattice is established.By mean of implication operation, implicative closure operators are defined on complete doubleresiduated lattices, some equivalent characterizations are given. The axiomatic systems of fuzzyrough upper and lower approximation operators based on regular commutative double residuatedlattices are investigated. |
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