代数结构入门:群、环、域、向量空间

抽象代数研究对象是代数结构(集合+一套运算规则),以本人的理解方式整理了代数结构基础内容,旨在弄清群、环、域等代数结构间的关系。(之前很认真整理了这部分内容,后来装双系统把草稿给丢了,这会重新整理)

1. 抽象代数

1.1 相关概念

数学的发展通常是先从简单的开始,然后不断放宽限制,推广到更一般化。从初等代数到抽象代数很好说明了这一点。

(1)算术(arithmetic)

算术(arithmetic)无疑是数学中最古老、最基础和最初等的部分。算术研究数的性质及其运算。把数和数的性质、数和数之间的四则运算在应用过程中的经验累积起来,并加以整理,就形成了最古老的一门数学——算术[1]。值得一提的是,算术运算不仅仅指加减乘除,还可以是百分比、平方根、取幂和对数;算法的对象包括自然数、整数、有理数和实数(兴许还包括复数);进制不仅仅是十进制,还可以是二进制、十六进制、六十进制。个人认为,算术的最大特点是关注具体数字。

(2)初等代数

用符号(成了变量)代替具体的数字,就可以得到更一般化(generalization)的等式,举例如下:

(2 + 3)^2 = 2^2 + 2 \times 2 \times 3 + 3^2 \\
(3 + 5)^2 = 3^2 + 2 \times 3 \times 5 + 5^2 \\
\Rightarrow\\
(a + b)^2 = a^2 + 2 \times a \times b + b^2

图1 从算术到代数的例子

初等代数(elementary algebra)是古老算术的推广与发展。在古代,算术积累了大量数量问题的解法,为寻求更系统、更普遍的求解各种数量关系方法,就产生了以解方程为中心的初等代数[2]。从实际问题的数量关系(即代数式:整式、分式、根式)、等量关系(或者不等式)列出列出方程或者方程组。方程(组)包括一元/二元一次方程(linear equations with one/two variable)、一元二次方程(quadratic equations)、指数和对数方程(exponential and logarithmic equations)、无理方程(radical equations)、线性方程组(system of linear equations)[3]

高等代数相对于初等代数而言,本质上是一个东西,只是更加系统(深度+广度)。

初等代数再进一步推广(generalization),那就是抽象代数了。初等代数与抽象代数的界限在于初等代数只考虑实数和复数代数结构。摘录维基百科词条Elementary algebra

Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

(3)抽象代数

抽象代数(abstract algebra)、近世代数、现代代数(modern algebra)指的都是同一个意思(甚至直接称为代数学)。抽象代数主要研究对象是代数结构,包括群、环、域、向量空间。

伽罗瓦(Évariste Galois, 1811-1832)是现代群论的创始人(与阿贝尔独立发明),他利用群的概念彻底解决了用根式求解代数方程的可能性问题[4](称为伽罗瓦理论),系统阐释了为何五次以上之方程式没有公式解,而四次以下有公式解[5],使代数学从解方程的科学转变为研究代数结构的科学,即把代数推广到抽象代数[4]

(4)线性代数

线性代数是抽象代数特殊的一类,其代数结构为:向量空间(vector spaces,也叫线性空间) + 线性变换(linear mappings)很容易将线性代数和矩阵理论等同起来,但其实是不一样的,讨论线性变换是基于选定一组基的前提下。摘抄mathoverflow上的一个回答(原文在这里):

When you talk about matrices, you’re allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another.

In linear algebra, however, you instead talk about linear transformations, which are not a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. However, when you’re given a linear transformation, you’re not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you’re only allowed to ask for things that don’t depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.

1.2 代数结构

既然抽象代数研究对象是代数结构(algebraic structure),那什么是代数结构呢。看了多个不同角度描述代数结构,如百度百科代数:代数是研究数、数量、关系与结构的数学分支。还是觉得《[转]MIT牛人解说数学体系》中的描述最深入浅出,如下:

代数主要研究的是运算规则。一门代数, 其实都是从某种具体的运算体系中抽象出一些基本规则,建立一个公理体系,然后在这基础上进行研究。一个集合再加上一套运算规则,就构成一个代数结构[1](想想计算机的数据结构:数据+操作)

1.3 初等代数–>抽象代数

抽象代数将初等代数的一些概念延伸。

(1)数 –> 集合

集合在朴素集合论(naive set theory)和公理化集合论(axiomatic set theory)的定义是不一样的,前者指由一些元素组成;后者指具有某种特定性质事物的总体。还是看维基英文词条吧[7]

Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property specific for the set.

(2)+ –> 二元运算

加号+被抽象为二元运算*(binary operation),对两个元素作二元运算,得到的新元素仍然属于该集合,这叫封闭性(closure)。实际上,加减乘除都叫二元运算(二元指的是两个操作数)。原文如下[7]

The notion of addition (+) is abstracted to give a binary operation, ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, ab is another element in the set; this condition is called closure.

Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

(3)0/1 –> 单位元

0和1被抽象成单位元(identity elements),0为加法单位元,1为乘法单位元。单位元是集合的一个特殊元素(跟二元运算有关),满足单位元与其他元素相结合时,不改变该元素,即满足ae = aea = a。可见,单位元取决于元素与二元运算,如矩阵的加法单位元是零矩阵,矩阵的乘法单位元是单位矩阵。值得注意的是,有些集合不存在单位元,如正整数集合(the set of positive natural numbers)没有加法单位元(no identity element for addition)。维基百科原文如下:

The numbers zero and one are abstracted to give the notion of an identity element for an operation. An identity element is a special type of element of a set with respect to a binary operation on that set. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a. Not all sets and operator combinations have an identity element.

(4)负数 –> 逆元素

负数推广到逆元素(inverse element),对于加法,a的逆元素是-a;对于乘法,a的逆元素是倒数a−1。直观地说,逆元可以撤销操作,如加了一个数a,再加上该数的逆元-a(相当于撤消操作),结果还是一样。维基百科原文如下:

The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written −a, and for multiplication the inverse is written a−1. A general two-sided inverse element (left inverse + right inverse) a−1 satisfies the property that a ∗ a−1 = 1 and a−1 ∗ a = 1.

The idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication. The intuition is of an element that can ‘undo’ the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.

(5)结合律

结合律(Associative property)是某些二元运算的性质,有些二元运算没有结合律(如减法、除法、八元数)。原文如下:

Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. In general, this becomes (ab) ∗ c = a ∗ (bc). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

(6)交换律

交换律(Commutative property),改变二元运算符两边的元素不影响结果。并不是所有二次元运算都满足交换律(如矩阵的乘法)。维基百科原文如下:

A binary operation is commutative if changing the order of the operands does not change the result. Addition and multiplication of real numbers are both commutative. In general, this becomes ab = ba.

This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

2. Group-like

代数结构(R, *),二元运算根据封闭性、单位元、逆元、结合律、交换律,可以归纳成不同的群。本节介绍的group-like,从最不严格到严格(依次添加限制条件),其关系图如下:

image

图1 群之间的关系

维基百科有一张表,给出更详细的group-like间的关系,如下:

image

图2 Group-like structures (source from here)

2.1 原群

原群(magma)是一种基本的代数结构,只要满足两元素作二元运算得到新元素仍属于该集合,即封闭性。维基百科原文如下:

A magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M. The binary operation must be closed by definition but no other properties are imposed.

2.2 半群

半群(Semigroup),满足结合律(associative property)的代数结构。V=<S,* >,其中二元运算*是可结合的,即(a*b)*c=a*(b*c),则称V是半群。维基百科原文如下:

A semigroup is an algebraic structure consisting of a set together with an associative binary operation.

A semigroup generalizes a monoid in that a semigroup need not have an identity element. It also (originally) generalized a group (a monoid with all inverses) in that no element had to have an inverse, thus the name semigroup.

2.3 幺半群

幺半群(monoid)在半群的基础上,还需要满足有一个单位元。维基百科原文如下:

A monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity.

2.4 群

群(group)是两个元素作二元运算得到的一个新元素,需要满足群公理(group axioms),即:

  • 封闭性:ab is another element in the set
  • 结合律:(a ∗ b) ∗ c = a ∗ (b ∗ c)
  • 单位元:a ∗ e = a and e ∗ a = a
  • 逆  元:加法的逆元为-a,乘法的逆元为倒数1/a,… (对于所有元素)

如整数集合,二次元运算为加法就是一个群(封闭性是显然的,加法满足结合律,单位元为0,逆元取相反数-a)。维基百科原文如下:

A group is a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility.

One of the most familiar examples of a group is the set of integers together with the addition operation

2.5 阿贝尔群(交换群)

阿贝尔群(Abelian Group)在群的基础上,还需满足交换律。如整数集合和加法运算,(Z,+),是一个阿贝尔群。

  • 群公理:见2.4 群。
  • 交换律:a + b = b + a

维基百科原文如下:

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity).

3. 环论

环在交换群基础上,进一步限制条件。环、交换环、域间的关系如下:

image

图3 环、交换环、域间的关系

维基百科有一张表从不同角度呈现这三者的关系,如下:

image

图4 Ring-like structures (source from here)

3.1 环

环(ring)在阿贝尔群(也叫交换群)的基础上,添加一种二元运算·(虽叫乘法,但不同于初等代数的乘法)。一个代数结构是环(R, +, ·),需要满足环公理(ring axioms),如(Z,+, ⋅)。环公理如下:

(1)(R, +)是交换群

  • 封闭性:a + b is another element in the set
  • 结合律:(a + b) + c = a + (b + c)
  • 单位元:加法的单位元为0,a + 0 = a and 0 + a = a
  • 逆  元:加法的逆元为-a,a + (−a) = (−a) + a = 0 (对于所有元素)
  • 交换律:a + b = b + a

(2)(R, ·)是幺半群

  • 结合律:(ab) ⋅ c = a ⋅ (bc)
  • 单位元:乘法的单位元为1,a ⋅ 1 = a and 1 ⋅ a = a

(3)乘法对加法满足分配律Multiplication distributes over addition

  • a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) for all a, b, c in R (left distributivity)
  • (b + c) ⋅ a = (b ⋅ a) + (c ⋅ a) for all a, b, c in R (right distributivity)

维基百科原文如下:

A ring is an abelian group with a second binary operation (The abelian group operation is called “addition” and the second binary operation is called “multiplication” in analogy with the integers) that is distributive over addition and is associative.

One familiar example of a ring is the set of integers. The integers are a commutative ring, since a times b is equal to b times a. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of square matrices of the same size. Finally, a field is a commutative ring in which one can divide by any nonzero element: an example is the field of real numbers.

3.2 交换环

交换环(commutative ring)在环的基础上,二元运算乘法还满足交换律

A commutative ring is a ring in which the multiplication operation is commutative

3.3 整环

整环(integral domain)在交换环的基础上,并满足没有零因子(如此,集合内任意两个元素乘积均不等于0)。维基百科原文如下:

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain the cancellation property holds for multiplication by a nonzero element a, that is, if a ≠ 0, an equalityab = ac implies b = c.

4. 域

域(Field)在交换环的基础上,还增加了二元运算除法,要求元素(除零以外)可以作除法运算,即每个非零的元素都要有乘法逆元。由此可见,域是一种可以进行加减乘除(除0以外)的代数结构,是数域与四则运算的推广。整数集合,不存在乘法逆元(1/3不是整数),所以整数集合不是域。有理数、实数、复数可以形成域,分别叫有理数域、实数域、复数域。域的几种定义,直接看维基百科英文吧:

a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element,

or equivalently a ring whose nonzero elements form an abelian group under multiplication.

As such it is an algebraic structure with notions of addition,subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law.

从有限域到交换环一些代数结构的从属关系如下:

Commutative ringsintegral domainsintegrally closed domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfieldsfinite fields.

5. 向量空间

向量空间(vector space)是一些向量的集合。最熟悉的例子是几何向量或矢量(Euclidean vectors, geometric vector, spatial vector),表示具有大小和方向的对象,如\overrightarrow{AB}.,矢量可以做加法(addition)和乘法(scalar multiplication)运算,举例如下:

图5 Vector addition and scalar multiplication (source from here)

其他例子,还包括坐标空间(Coordinate spaces)、复数、函数空间(Function spaces)、线性方程组(linear equations)。详情可查阅维基百科词条:Examples of vector spaces.

5.1 8个公理

摘抄维基百科Vector space部分内容如下:

A vector space is a collection of objects called vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars in this context. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

给定域F,向量空间V记为F-向量空间。其二元运算:

  • 向量加法:+ : V × V → V 记作 v + w, ∃ v, w ∈ V
  • 标量乘法:·: F × V → V 记作 a v, ∃a ∈ F 且 v ∈ V

并且满足如下8条公理[10]

  • 向量加法结合律:u + (v + w) = (u + v) + w
  • 向量加法的单位元:V存在零向量的0,∀ v ∈ V , v + 0 = v
  • 向量加法的逆元素:∀v∈V, ∃w∈V,使得 v + w = 0
  • 向量加法交换律:v + w = w + v
  • 标量乘法与域乘法兼容性(compatibility): a(b v) = (ab)v
  • 标量乘法有单位元: 1 v = v, 1指域F的乘法单位元
  • 标量乘法对于向量加法满足分配律:a(v + w) = a v + a w
  • 标量乘法对于域加法满足分配律: (a + b)v = a v + b v

另,若F是实数域ℝ,则V称为实数向量空间;若F是复数域ℂ,则V称为复数向量空间;若F是有限域,则V称为有限域向量空间。

6. 模

模(module)是对向量空间的推广,将标量需为域(向量空间)推广到任意环(模)。维基百科module部分原文如下:

A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary ring.

7. 代数(环论)

代数(algebra)将algebra over a field中的域推广到交换环。维基百科原文如下:

An algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R.

An algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars. Such an algebra is called here a unital associative algebra for clarity, because there are also nonassociative algebras.

8. 格

格(lattice)是任意两个元素都有上确界和下确界的偏序集合。

A lattice is a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).

An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

9. 总结

是时候,祭出这张图了,图片来源于这里。YouTube还有一段小视频讲这张图的,在这里

图6 Mathematical Structures/Objects

参考资料:
[1]百度百科词条:算术
[2]百度百科词条:初等代数
[3]Wikipedia: Elementary algebra
[4]百度百科词条:抽象代数
[5]百度百科词条:埃瓦里斯特·伽罗瓦
[6][转]MIT牛人解说数学体系
[7]Wikipedia: Algebra#Abstract_algebra
[8]博文系列:明日枯荷包:数学闲话
[9]Wikipedia: Vector space
[10]百度百科词条:向量空间

发表评论

电子邮件地址不会被公开。 必填项已用*标注

14 thoughts on “代数结构入门:群、环、域、向量空间