Roughness in quotient groups

TL;DR

This paper explores the application of rough set theory to quotient groups, defining lower and upper approximations, and investigates their algebraic properties and relationships, including conditions for normality and homomorphisms.

Contribution

It introduces the concepts of lower and upper approximations in quotient groups and analyzes their properties, including conditions for normality and the development of homomorphisms.

Findings

Lower approximation can be a normal subgroup under certain conditions.

Upper approximation generally does not form a normal subgroup.

Homomorphisms between lower approximations are established.

Abstract

The theory of rough sets was firstly introduced by Pawlak (see \cite{p}). Many Mathematician has been studied the relations between rough sets and algebraic systems such as groups, rings and modules. In this paper we will introduce the lower and upper approximations in a quotient group. We will discuss several properties of the lower and upper approximations. Moreover under some additional assumptions we are able to show that the lower approximation is a normal subgroup of the quotient group but this property fails for the upper approximation. At the end we will develop several homomorphisms between lower approximations.