Algebraic Structure Multi-level Classification System: Axiomatic, Extensible, and Periodic Table-style Prediction

This paper establishes a multi-level classification system for algebraic structures based on operation symbols and their axiom systems. The system uses the number of operations, sequence types, arities and reversibility, generation rules, interaction axioms, identities, and concrete instances as successive refinement criteria, forming an arbitrarily extensible classification tree. Each level is equipped with corresponding axiom systems and fundamental theorems, enabling any algebraic structure to be uniquely placed into a specific node of the tree. Conversely, any node’s parameter combination mechanically generates an axiom system and predicts algebraic structures not yet studied. The system possesses uniformity, completeness, and extensibility, analogous to the periodic table of chemical elements, and can be used for systematic discovery and construction of new algebraic systems. This paper provides formal definitions, construction methods, fundamental theorems, and multiple examples, demonstrating how existing mathematical branches are embedded into the system and how axiom systems and fundamental theorems for vacant parameters are built. All theorems are given rigorous proofs (general theorems at least 4 steps, important theorems at least 8 steps), and all predictions are equipped with complete axiom systems and existence constructions, transforming mathematical development from random discovery to fill-in-the-blank construction.