Abstract
From equivalence relation RB δ on discourse domain U , we can derive equivalence relation Rδ on the attribute set A . From equivalence relation Rδ on discourse domain A , we can derive a congruence relation on the attribute power set P (A ) and establish an object dependent space. And then,we discuss the reduction method of fuzzy information system on object dependent space. At last ,the example in this paper demonstrates the feasibility and effectiveness of the reduction method based on the congruence relation Tδ providing an insight into the link between equivalence relation and congruence relation of dependent spaces in the rough set. In this way, the paper can provide powerful theoritical support to the combined using of reduction method, so it is of certain practical value.
Highlights
Definition 1 Suppose that U, A, F is a fuzzy information system,where U is a limited object discourse domain, A is a limited condition attribute set, F is a mapping set
Note: The above deduction demonstrates that the closure operator on limited semi-lattice and congruence relation can determine each other
It is feasible to discuss the issue of reducing attribute of fuzzy information system based on the attribute dependent space A,T G A C (T G) built on attribute set A
Summary
Definition 1 Suppose that U , A, F is a fuzzy information system,where U is a limited object discourse domain, A is a limited condition attribute set, F is a mapping set. R~ is a fuzzy equivalence relation on the discourse domain U , R~ :U uU o >0,1@. Suppose that it has the following attributes: 1) R~ xi , xi 1; 2) R~ xi , x j R~ x j , xi ; 3). Definition 3 Suppose that S,q is a semi-lattice, C : S o S is a closure operator, if the following conditions hold for x, y S : 1) x d C x ; 2) x d y CxdCy ; 3) CCxCx. Theorem 2 Let RG be an equivalence relation on, P(U ) is a division of P U (discourse power set ). For Y C C RG X Y P U RG Y C RG X z I , from the definition of operator we can derive Y X z I Y C RG X , from the arbitrariness of Y we can derive C RG X C C RG X
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