离散数学课件(英文版)----Semigroup.ppt

1、Algebral systems and groups，leture 13 discrete mathematical structures，algebral operations，Function : anb is called an-0.5 * 0.7=0.85，closessions，for anyoperation : anb，if ba，then it is said that a is closed with respect to . or，We，association，operation defined on the set a is associative if and onl

2、y if : for anyx，y，z a，(xy) z=x(，Commutation，operation Defined on the set a is associative if and only if : for anyx，y a，xy=yx if is commumum z a，x (yz)=(xy) (xz) (exactly speaking，this is the first distributive property)，Identity of an algebic it is not that every algebraic system has its identity e

3、lement。Left identity and right identity，El is called a Left identity of an algebbraic system s，if and only if : for anyxs，elx=x rightCan be defined similari，more about identity，For any algebic system s : there may or may not be left or right identity . there may be inverse，(inverse can be discussed

4、for those system with identity)for a given element x in the system s，if there is some element x in the system s satisfying that : xx *=x * x=1s，Then x * is called an inverse of x，de notes as x-1。an example about inverse，note :*，a，b，c， and the right inverse is x : x=x1s=x(xx)=(xx)x =1sx =x if every e

5、lement of s has a left inverse for any a in s，let b is a left inverse of a，Let c is a left inverse of b，then : ab=(1sa) b=(CB) a) b=() 0 x=0 an element z in s is The zero element of s if and only if，for anyxs，zx=xz=Z. it is easy to prove that，if existities de notation : 0s，or simply，0，but remember t

6、hat it is not that 0 。a system may not have a zero element。An example，define a binary operation on the set of real number，using arithmetic operations as follows 3360 for any real number)inverse of x(x1): x/(x-1)(note : 1 has no inverse！)，Properties and Operation Table，Commutation : symmetric matric

7、matrix identity : there is a row/Column identical to the title row，seigroup，axiom of seigroup association an example(1，2，*)，* defined as follows : for anyx，y1，2，Niels Abel(1802-1829): genius and poverty，Abel的第一次雄心勃勃的冒险是尝试解决一般的五次方程式。失败给了他很有益的打击。那使他怀疑代数解法是否可行。他证明了不可侵犯。那时他大概19岁。亚伯关于非常广泛的一类超越函数一般性质的论文提交巴黎科学院。这就是勒让德后来用荷鲁斯的话描述“永恒的纪念碑”的职业。埃尔米特说：“他对数学家们保持了足够的忙碌500年。”这是现代数学最高峰的业绩。来自贝尔的数学精英)的一位评论家勒尚德74岁，这是一篇很难读的论文，另一位评委39岁的柯西，把论文从以自我为中心的顶峰带回了家，忘得一干二净。四年后，当柯西终于把它翻过来的时候，亚伯不再死了。作为补偿，科学院让亚伯和雅可比亚一起获得了1830年的数学大奖。mono id，axioms of the system : association，identity an example : let * be matrix multi plic