# Rough Algebraic Structures Corresponding to Ring Theory

Chapter
Part of the Trends in Mathematics book series (TM)

## Abstract

The concept of rough set was originally proposed by Pawlak in 1982. Since then the subject has been investigated in many papers. Some authors studied algebraic properties of rough sets. The lattice theoretical approach has been suggested by Iwinski. Pomykala and Pomykala showed that the set of rough sets forms a Stone algebra. Comer presented an interesting discussion of rough sets and various algebras related to the study of algebraic logic, such as Stone algebras and relation algebras. It is a natural question to ask what does happen if we substitute an algebraic structure instead of the universe set. Biswas and Nanda introduced the notion of rough subgroups. Kuroki introduced the notion of a rough ideal in a semigroup. Kuroki and Wang gave some properties of the lower and upper approximations with respect to the normal subgroups. Also, Kuroki and Mordeson studied the structure of rough sets and rough groups. Jun applied the rough set theory to BCK-algebras. The present author applied the concept of approximation spaces in ring theory, module theory and algebraic hyperstructures. A key notion in Pawlak rough set model is an equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. An equivalence relation is sometimes difficult to be obtained in real-world problems due to the vagueness and incompleteness of human knowledge. From this point of view, the author introduced the concept of lower inverse and upper inverse of a set under a set-valued map, which is a generalization of the lower and upper approximations. Using this the concept of a set-valued homomorphism for groups, rings, modules and lattices was introduced. The concept of uniform set-valued homomorphism was introduced and it was shown by the present author that every set-valued homomorphism is uniform. The overall aim of this chapter is to present an introduction to some of these results, methods and ideas about rough algebraic structures. Most of the focus will be on rough rings and their generalizations.

16Y99 03E72

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