Binary extended theta operation of binary soft sets

Binary soft set theory, first introduced by Açıkgöz and Taş in 2016, has become widely accepted as a technique for addressing and modeling uncertainty. Numerous theoretical and practical problems have been solved using this approach. Scholars have shown sustained interest in the theory's core concepts and operations since its inception. In this study, we propose the binary extended theta operation, a special binary soft set operation, and provide a thorough analysis of its basic algebraic features. We also study the distribution of this operation over certain types of binary soft set operations. By considering its algebraic properties and distribution rules, we show that, when combined with specific binary soft set operations, the binary extended theta operation forms many important algebraic structures within the collection of binary soft sets over the universe under certain conditions. The fundamental conceptual difference between the proposed binary extended theta operation and existing binary extended operations in the literature is that unlike approaches based on positive information aggregation, the theta operation systematically extracts negative information through common parameters and offers a unique and complementary tool, particularly for decision problems requiring reliable elimination, risk exclusion, and error detection. Further applications, including cryptology and decision-making, rely on operations of binary soft sets, making this theoretical subject essential from both theoretical and practical perspectives.