Algebraic structures on nearness approximation spaces

This study which is designed as a philosophy doctoral thesis covers four chapters. In the first chapter, some basic concepts such as rough sets, near sets, fundamental approximation spaces and nearness approximation spaces were given for the rest of the thesis that readers can easily understand. In the last section of this chapter, description based set operators were also given. Second chapter is devoted to the nearness semigroups. In this chapter, complete indiscernibility relation and some properties of lower and upper approximations on nearness approximation spaces were obtained. Nearness semigroups and nearness ideals of nearness semigroups were studied considering the upper approximation of a nonempty set on nearness approximation spaces. In the third chapter, nearness groups were investigated which are one of the algebraic stuructures that has only one binary operation on nearness approximation space. In the first section of this chapter, nearness groups were studied with some examples. Some basic properties of nearness subgroups and nearness normal subgroups were given. Weak cosets were introduced taking into consideration nearness groups and necessary conditions were investigated for the existence of the nearness groups of weak cosets without use of nearness normal subgroups. In the last section of this chapter, nearness group homomorphisms and some results related to the nearness group homomorphisms were given. In the last chapter, nearness rings were investigated which are the nearness algebraic sturucture that have two binary operations. Also in this chapter, nearness rings and nearness ideals of nearness rings were studied by giving some examples. Weak equivalence relation on nearness ring and weak cosets that determined by weak equivalence relation on nearness ring were defined. Necessary conditions were obtained for the existence of the nearness rings of weak cosets without using nearness ideals. In the last section of this chapter, nearness ring homomorphisms and some results related to the nearness ring homomorphisms were investigated. Finally, concept of restricted nearness homomorphism was introduced and fundamental nearness ring homomorphism theorem was obtained by using this concept.