数据系统的代数框架：面向数据组织与编码的代数结构分类与优化
An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding
- Pontificia Universidad Católica Madre y Maestra (PUCMM)（圣母主教师大学）
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AI总结：
本文提出一种基于群论与编码理论的代数框架，建立有限代数结构编码能力的公理分类定理，构造特殊编码并提出优化泛函，为数据系统建模与优化提供新方法。
AI中文摘要：
现代数据系统通常通过计算方法和信息论方法进行研究，而其代数特性在很大程度上仍未被探索。本文引入一种数学框架，利用群论和编码理论中的代数结构对数据系统进行建模。核心成果是一个公理分类器：有限代数结构（群、环或域）的编码理论能力由其公理签名（即它满足的代数公理集合）决定，每个公理如同一个门，使能特定能力（逆元使能代数汉明度量；交换性使能通过商群进行伴随式译码；域结构使能通过多项式求值实现MDS码）。本文呈现三类结果：已证明的成果包括公理分类定理；ACID事务在顺序复合下构成一个幺半群（而非群）；保持模式的变换构成一个有限群，其轨道通过Burnside引理计数，可实现等价配置的精确去重；数据系统的每个完整性保持双射在有限群作用下定义一个可代数化的等价类。已演示的成果包括在$\boldsymbol{\text{F}}_5$、$\boldsymbol{\text{Z}}_4$上的显式码构造（产生经典$\boldsymbol{\text{F}}_q$-线性理论无法实现的码），以及在$\boldsymbol{(2^U, \triangle)}$上的构造（产生面向集合值数据的群论异常检测）。已提出的成果包括编码效率度量$\text{Ef}$和带可调权重的优化泛函$\boldsymbol{\text{\u03A6}}$，其诱导的排序经计算验证与所研究结构的公理分类一致。
英文摘要：
Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic structures drawn from group theory and coding theory. The central result is an axiomatic classifier: the coding-theoretic capability of a finite algebraic structure (group, ring, or field) is shown to be determined by its axiom signature (the set of algebraic axioms it satisfies), with each axiom acting as a gate that enables a specific capability (inverses enable the algebraic Hamming metric; commutativity enables syndrome decoding via quotient groups; field structure enables MDS codes via polynomial evaluation). Three types of results are presented. Proved: the axiomatic classification theorem; that ACID transactions form a monoid under sequential composition (not a group); that schema-preserving transformations form a finite group whose orbits, counted by Burnside's lemma, yield exact deduplication of equivalent configurations; and that every integrity-preserving bijection of a data system defines an algebraizable equivalence class under a finite group action. Demonstrated: explicit code constructions over $\mathbb{F}_5$, $\mathbb{Z}_4$ (yielding codes inaccessible to classical $\mathbb{F}_q$-linear theory), and $(2^U, \triangle)$ (yielding group-theoretic anomaly detection for set-valued data). Proposed: an encoding efficiency measure Ef and an optimization functional $Φ$ with tunable weights, whose induced ranking is verified computationally to be consistent with the axiomatic classification for the structures studied.
