A thorough analysis of soft products from certain types of subsethood perspective: Part 1

Since its seminal introduction by Molodtsov in 1999, soft set theory has evolved into a pivotal mathematical paradigm for the formal representation and management of uncertainty, offering a flexible and conceptually robust alternative to classical approaches. Its methodological versatility and abstract generality have fostered an expanding body of theoretical inquiry and practical deployment across diverse scientific domains. Central to the structural framework of soft set theory are its operations, which constitute the algebraic backbone of the theory and have consequently remained an enduring focus of research interest. Within this operational landscape, several soft product variants of the soft sets have been introduced; however, the extant literature provides only fragmentary examinations of their interrelations as regard soft subsets. The present study seeks to rectify this deficiency by delivering a comprehensive and rigorous analysis of this issue, with particular reference to the fundamental conditions which soft product, starting with intersection and union, is is soft subset of another soft product, and when these soft products are equal. That is, the study has been further expanded to provide a deeper analysis of which soft product is a subset of another and the circumstances under which they become equal. In pursuing this objective, the investigation elucidates a series of structural and distributive properties that not only reinforce the algebraic coherence of the soft product variants of the soft sets but also reveal their potential to generate rich algebraic configurations. Thus, the present theoretical exposition advances the foundational algebraic framework of soft set theory and contributes to a deeper understanding of its operational architecture. Beyond its intrinsic theoretical relevance, this deepened analysis is of substantial theoretical significance, as it elucidates the hierarchical and structural relationships among distinct soft products, thereby enhancing the algebraic understanding of the internal organization of soft set theory and facilitating the identification of operational dependencies and equivalences.