Matrix Science Mathematic (MSMK)

Soft set theory provides a mathematically robust and algebraically versatile framework for modeling systems characterized by epistemic indeterminacy, vagueness, and parameter-dependent variability—features that pervade foundational inquiries in decision theory, engineering, economics, and the information sciences. At the heart of this formalism lies an extensive suite of algebraic operations and binary product constructs that collectively confer a rich internal structure upon the universe of soft sets, capable of faithfully representing intricate parametric interrelations. Within this conceptual setting, we introduce and rigorously investigate a novel soft product, referred to as the soft union–theta product, defined over soft sets whose parameter spaces are endowed with an intrinsic group-theoretic structure. The operation is meticulously axiomatized to ensure compatibility with generalized soft subsethood and equality relations, thereby preserving the formal algebraic integrity of the resulting system. A comprehensive algebraic analysis is undertaken to characterize the operation’s fundamental properties—including closure, associativity, commutativity, idempotency, and interactions with identity and absorbing elements—as well as its behavior in relation to the null and absolute soft sets. In parallel, the proposed product is analytically juxtaposed with existing soft binary operations within the stratified hierarchy of soft subset classifications, offering deeper insights into their relative expressive capacities and mutual structural coherence. Our results affirm that the soft union–theta product respects the algebraic constraints imposed by group-parameterized domains while generating a coherent and structurally consistent algebraic system over the space of soft sets. Two core algebraic contributions emerge from this study: (i) the integration of this product fortifies the internal operational harmony of soft set theory by embedding it into an axiomatic framework that preserves and extends fundamental algebraic behaviors; and (ii) the operation lays the groundwork for a generalized soft group theory, wherein soft sets over group-based parameter domains replicate the axiomatic signatures of classical group structures under suitably defined soft operations. By addressing the critical need for algebraic operations grounded in semantically meaningful and structurally sound axioms, this work significantly advances the algebraic unification and generalization of soft set theory. Beyond its theoretical depth, the proposed operation enables the construction of abstract algebra-driven soft computational models, with direct implications for multi-criteria decision-making, algebraic classification mechanisms, and uncertainty-sensitive data analysis over group-structured semantic spaces. Consequently, the algebraic apparatus formulated herein not only extends the theoretical boundaries of soft algebra but also solidifies its foundational relevance in both abstract mathematical logic and applied analytical domains.