A Unified Mathematical Framework for Query Algebras Across Heterogeneous Data Paradigms

This paper presents a comprehensive, rigorous mathematical framework that unifies operations across transaction processing, text retrieval, and vector search paradigms within a single algebraic structure. By establishing posting lists as a universal abstraction with well-defined algebraic properties, we develop a theoretical foundation that preserves the expressivity of each paradigm while enabling cross-paradigm operations. We prove that these operations form a complete Boolean algebra and demonstrate how this structure supports query optimization through lattice theory. We extend this framework to incorporate join operations across paradigms, providing formal definitions and analyzing their computational properties. Further, we address limitations of the base framework by introducing formal models for aggregations and hierarchical data structures. Through category theory and information theory, we establish the soundness of our extensions and explore fundamental computational boundaries. The mathematical formalization includes precise definitions of operators, transformation rules with formal proofs, optimization techniques, and theoretical limitations. This unified framework provides a solid foundation for next-generation data systems that must seamlessly operate across previously isolated paradigms.