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The algebraic structure of rough linear approximation space
LIU Yamei, MA Yingcang, LU Wenxia, CHEN Yanyan
School of Science,Xi'an Polytechnic University,Xi'an 710048,China
Abstract: Focusing on the description of the algebraic structure on the upper(lower)approximation,and according to the properties of the upper(lower)approximation in rough linear space,two new sets are presented,and the inclusion relation of the upper approximation's union and the lower approximation's intersection are improved,deriving the equation expressions of the upper approximation's union and the lower approximation's intersection. Moreover,the rough linear approximation space is proposed and the intersection,union and complementary operations are introduced in the rough linear approximation space. Finally,it has been proven that the rough linear approximation space is Boolean algebra on the intersection,union and complementary operations. This paper enriches the combination of linear space and rough set research.
Key words: rough linear approximation space     linear subspace     congruence     upper and lower approximation     Boolean algebra     algebraic structure

1 基本概念

1)ρWV上的一个同余关系

2)∀αV，同余类则可将记为是全体同余类的集合。

1)

2)

3)

4)若，则有,

5)

6)

2 集合的交(并)的上(下)近似的等式刻画

;

,

1)

2)

，则有,或，则。若，则，所以

，有

，则有，若，则有，若，则，则有

2)此命题等价为

，有

，则有，若，则有，所以，则。对，则有，即

，则有；若，则

3 粗糙线性近似空间及其代数结构

1)

2),

3),

4),

2)

1)交换律：

2)结合律：

3)分配律：

4)幂等律：

5)0-1律：

6)互补律：

7)对偶律：

1)由PX(Y)和QX(Y)定义可以看出PX(Y)=PY(X)，QX(Y)=QY(X)。

2)

3)

4)

5)

6)

7)要证，只需证

4 结束语

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DOI: 10.3969/j.issn.1673-4785.201307016

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#### 文章信息

LIU Yamei, MA Yingcang, LU Wenxia, CHEN Yanyan

The algebraic structure of rough linear approximation space

CAAI Transactions on Intelligent Systems, 2014, 9(2): 224-228
http://dx.doi.org/10.3969/j.issn.1673-4785.201307016