Irredundant Coverings, Tolerances, and Related Algebras
This chapter deals with rough approximations defined by tolerance relations that represent similarities between the elements of a given universe of discourse. We consider especially tolerances induced by irredundant coverings of the universe U. This is natural in view of Pawlak’s original theory of rough sets defined by equivalence relations: any equivalence E on U is induced by the partition U∕E of U into equivalence classes, and U∕E is a special irredundant covering of U in which the blocks are disjoint. Here equivalence classes are replaced by tolerance blocks which are maximal sets in which all elements are similar to each other. The blocks of a tolerance R on U always form a covering of U which induces R, but this covering is not necessarily irredundant and its blocks may intersect. In this chapter we consider the semantics of tolerances in rough sets, and in particular the algebraic structures formed by the rough approximations and rough sets defined by different types of tolerances.
We would like to thank our second reader Magnus Steinby for the effort he has put in to provide expert views on our original manuscript. His many valuable comments and suggestions have veritably improved the quality of this work.
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