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半群 (Semigroup)

An algebraic structure (G, *) is said to be a semigroup. If the binary operation * is associated in G i.e. if (a*b) *c = a *(b*c) a,b,c e G. For example, the set of N of all natural number is semigroup with respect to the operation of addition of natural number.

代数结构(G，*)被称为半群。 如果二进制运算*与G关联，即(a * b)* c = a *(b * c)a，b，ce G。 例如，相对于自然数的加法运算，所有自然数的N的集合是半群。

Obviously, addition is an associative operation on N. similarly, the algebraic structure (N, .)(I, +) and (R, +) are also semigroup.

显然，加法是对N的关联运算。 同样，代数结构(N，。)(I，+)和(R，+)也是半群。

单体 (Monoid)

A group which shows property of an identity element with respect to the operation * is called a monoid. In other words, we can say that an algebraic system (M,*) is called a monoid if x, y, z E M.

显示关于操作*的标识元素的属性的组称为monoid。 换句话说，如果x，y，z EM ，我们可以说一个代数系统(M，*)被称为一个等式 。

(x *y) * z = x * (y * z)

(x * y)* z = x *(y * z)

And there exists an elements e E M such that for any x E M

并且存在一个元素EM ，对于任何x EM

e * x = x * e = x where e is called identity element.

e * x = x * e = x其中e称为身份元素。

关闭属性 (Closure property)

The operation + is closed since the sum of two natural number is a natural number.

由于两个自然数之和是自然数，所以运算+是闭合的。

关联财产 (Associative property)

The operation + is an associative property since we have (a+b) + c = a + (b+c) a, b, c E I.

由于我们具有(a + b)+ c = a +(b + c)a，b，c EI，因此运算+是一种关联性质。

身分识别 (Identity)

There exist an identity element in a set I with respect to the operation +. The element 0 is an identity element with respect to the operation since the operation + is a closed, associative and there exists an identity. Since the operation + is a closed associative and there exists an identity. Hence the algebraic system ( I, +) is a monoid.

关于操作+ ，在集合I中存在一个标识元素。 元素0是关于操作的标识元素，因为操作+是封闭的，关联的并且存在一个标识。 由于操作+是封闭的关联，因此存在一个标识。 因此，代数系统(I，+)是一个齐半群 。

组 (Group)

A system consisting of a non-empty set G of element a, b, c etc with the operation is said to be group provided the following postulates are satisfied:

如果满足以下假设，则一个由元素a，b，c等组成的非空集G组成的系统将被称为组。

1. Closure property

1.关闭属性

    For all a, b E G => a, b E G
i.e G is closed under the operation ‘.’

2. Associativity

2.关联性

    (a,b).c = a.(b.c) a, b, c E G.
i.e the binary operation ‘.’ Over g is associative.

3. Existence of identity

3.身份的存在

    There exits an unique element in G. Such that e.a = a = a.e
for every a E G. This  element e is called the identity.

4. Existence of inverse

4.逆的存在

    For each a E G , there exists an element a^-1 E G
such that a. a^-1 = e = a^-1.a
the element a^-1 is called the inverse of a .

交换组 (Commutative Group)

A group G is said to be abelian or commutative if in addition to the above four postulates the following postulate is also satisfied.

如果除上述四个假设外，还满足以下假设，则称G组为阿贝尔或交换性的。

5. Commutativity

5.可交换性

    a.b = b.a for every a, b E G.

循环群 (Cyclic Group)

A group G is called cyclic. If for some aEG, every element xEG is of the form a^n. where n is some integer. Symbolically we write G = {a^n : n E I}. The single element a is called a generator of G and as the cyclic group is generated by a single element, so the cyclic group is also called monogenic.

组G称为循环的。 如果对于某些aEG ，每个元素xEG的形式都是a ^ n 。 其中n是一些整数。 象征性地，我们写G = {a ^ n：n EI} 。 单个元素a称为G的生成器，并且由于环状基团是由单个元素生成的，因此环状基团也称为单基因 。

亚组 (Subgroup)

A non- empty subset H of a set group G is said to be a subgroup of G, if H is stable for the composition * and (H, *) is a group. The additive group of even integer is a subgroup of the additive group of all integer.

一组群G的一个非空真子集H被表示为G的一个子群，如果H是稳定该组合物*和(H，*)是一组。 偶数整数的加法组是所有整数的加法组的子组。

翻译自: https://www.includehelp.com/basics/group-theory-and-their-type-in-discrete mathematics.aspx

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