New type of extended operations of soft sets: Complementary extended intersection, gamma and star operation

Soft set theory is seen as an effective mathematical tool in solving problems involving uncertainty, and has been applied in many theoretical and practical areas since its introduction. The basis for the theory is soft set operations. In this context, in this thesis thesis, new kind of soft set operations called complementary extended soft set operations are defined in order to contribute to the theory. All the properties of these operations called complementary extended intersection, complementary extended gamma and complementary extended star, are examined in detail, the distributions of each operations over other soft set operations are investigated in order to obtain the relationship of the operations with other soft set operations and it is obbserved that the complementary extended intersection operation on a set of fixed parameter set form many well-known and important algebraic structures in classic algebra. The thesis study consists of six sections, and in the first section, a literature review is conducted on soft sets and algebraic structures mentioned in the thesis. In the second part, basic definitions and properties of set operations, soft sets and some algebraic structures are given. In the third, fourth and fifth sections, which constitute the original part of the thesis, the definitions of complementary extended intersection, complementary extended gamma and complementary extended star operations are given, respectively, the algebraic properties of the operations are examined, and by handling the distributions of these new soft set operations over other soft set operations, not only the relationships of these operations with other soft set operations are obtained, but also it is observed what the algebraic structures these operations form. In addition, since the intersection operation is an operation that also exists in the classical set, the properties of this classical set operation are compared with the properties of the complementary extended intersection operation. In a way of contributing to soft set theory and the classical algebra both, it is presented that the complementary extended intersection operation on the set of soft sets with a fixed-parameter set form a bounded semi-lattice, and that the complementary extended intersection operation, along with other soft set operations, are commutative, idempotent, unitary, non-zero semiring, commutative, idempotent, unitary hemiring, Boolean Algebra, De Morgan Algebra, Kleene Algebra, Stone Algebra and MV-algebra. In the conclusion,the results obtained in the thesis and their importance are discussed.