Wave-Point Algebra

Wave-Point Algebra is a novel algebraic framework that interprets mathematical structures through the waves associated with Wave-Points, rather than through isolated algebraic elements. In this approach, algebraic operations are reformulated as transformations acting on wave information, where interactions such as interference, resonance, projection, lifting, and composition become the fundamental mechanisms. Instead of replacing classical algebra, Wave-Point Algebra offers a new perspective that reinterprets groups, rings, fields, modules, categories, and other algebraic structures in terms of wave dynamics. By treating algebraic elements as Wave-Points carrying intrinsic phase information, the framework naturally connects algebra with the geometric and analytic aspects developed in Wave-Point Geometry and Wave-Point Analysis. This paper introduces the core concepts of Wave-Points, Wave Sets, and Wave Mappings, along with basic operators including the Interference Operator, Resonance Operator, Projection Operator, and Lift Operator. These notions provide a unified foundation for constructing various Wave-Point algebraic systems and serve as a bridge toward a fully wave-based mathematical language.