[TOC]
↗ Set Theory & Axiomatic Set Theory
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.
Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow, called axioms. Universal algebra and category theory provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures.
Algebraic methods were first studied in the ancient period to solve specific problems in fields like geometry. Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, topology, number theory, and calculus, and other fields of inquiry, like logic and the empirical sciences.
Highly Abstract and Universal (高度抽象和统一）
| 学科 | 内容 | 时间 |
|---|---|---|
| 算数 | 算术运算 | 几千年 |
| 小代数 | 一次方程、二次方程 | 1千年 |
| 大代数 | 高次方程、线性方程组 | 16-19世纪 |
| 高等代数 | 线性代数(向量代数、矩阵代数)、多项式代数等,涉及具体代数结构 | 19-20世纪 |
| 抽象代数 | 代数系统、公理+结构 | 20世纪20年代 |
| 泛代数 | 范畴 | 近几十年 |
Axiomatic System Construction & Structure Analysis (公理化体系的建立和结构分析) 公理化体系：欧几里得的平面几何公理；集合论的公理化体系 结构分析：集合+对应规则+公理 = 结构
- 序结构
- 代数结构
- 拓扑结构
- 测度结构
- 上述结构的复合结构（有序距离线形空间）等
Interdiscipline (学科交叉、领域交叉) 数学研究领域交叉
- 泛函分析、解析数论
- 代数拓扑、代数图论 确定性与非确定性交叉
- 随机微分方程 与其他应用学科交叉
- 模糊数学
- 运筹学