Rough Subset Based on Congruence in a Vector Space

The rough set theory proposed by Pawlak, is a generalization of the classical set theory. Many important algebraic structures are naturally endowed with two binary operations: addition and multiplication, for example, rings, groups and modules. A vector space is an algebraic structure with a binary operation and a multiplication by a scalar. This paper concerned a relationship between rough sets and vector spaces. We considered a vector space as an universal set, and assumed that the knowledge about objects should be restricted by a subspace. First, we discussed relationships between congruences and subspaces of a vector space. Then we defined a pair of rough approximation operators based on a subspace, and obtained some properties of lower (upper) approximation of non-empty subsets of the vector space. Some characterizations of the approximation operators are expressed, and some counter examples are given.