Restricted and extended plus operations for soft sets

Soft set theory has gained importance as an innovative approach for addressing uncertainty-related problems and modeling uncertainty since it was introduced by Molodtsov in 1999. It has several applications in both theoretical and practical contexts. Soft set operations, the basic concept of the theory, have been of interest to researchers since the theory's inception, and some restricted and extended soft set operations have been defined and studied with their properties. This paper introduces a new restricted and extended soft set operation, which we refer to as restricted plus and extended plus operations. We also analyze these operations' fundamental algebraic characteristics in depth. Additionally, these operations' distributions over several soft-set operations are examined. By considering the algebraic properties of the operation and its distribution rules, we show that extended plus operation and other types of soft sets form many important algebraic structures, including semiring and near semiring in the collection of soft sets over the universe. Since the primary notion of the theory is the operations of soft sets as they serve not only the decision-making processes as accurately as possible but also the foundation for various applications, including cryptology, this theoretical study is of great importance both theoretically and practically.