Molodtsov introduced the theory of soft sets, which can be seen as an effective mathematical tool to deal with uncertainties, since it is free from the difficulties that the usual theoretical approaches have troubled. In this paper, we apply the definitions proposed by Ali et al. [M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009), 1547-1553] to the concept of soft near- rings and substructures of soft near-rings, proposed by Atag?n and Sezgin [A. O. Atag?n and A. Sezgin, Soft Near-rings, submitted] and show them with illustrating examples. Moreover, we investigate the properties of idealistic soft near-rings with respect to the near-ring mappings and we show that the structure is preserved under the near-ring epimorphisms. Main purpose of this paper is to extend the study of soft near-rings from a theoretical aspect.
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 basic concept of the theory is soft set operations. In this context, in this paper, a new kind of soft set operations called complementary extended soft set operation is defined in order to contribute to the theory. The properties of the operation are examined in detail together with its distributions over other soft set operations in order to obtain the relationship between complementary extended intersection operation and the others. We demonstrate that the collection of soft sets over with a fixed parameter set, along with the complementary extended intersection operation and other certain types of soft sets, form many well-known and important algebraic structures in classical algebra, including semiring, hemiring, Boolean ring, Boolean Algebra, De Morgan Algebra, Kleene Algebra, and Stone Algebra.