一、群論

幺半群(monoid)

之前看老師講的叫Abelian monoid（阿貝爾幺半群），但是搜不到。

A monoid is a set closed under an associative binary operation and has an identity element? such that?.

Binary operation(二元運(yùn)算): f(x, y) an operation between two quantities or expressions x and y.

An binary operation on a nonempty set? is a map of?.such that:

, "abelian commutative";

, "abelian associative";

, e is the "identity element".同上面的?.

常用的二元運(yùn)算有：addition , substraction , multiplication , division .

不同于群（group），幺半群中的元素不一定有逆。

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群(group)

也叫Abelian group.在 monoid 的基礎(chǔ)上多了一個(gè)條件:

Every element has an inverse:?, where e is the identity element.

例子：

Every vector space? is Abelian group.?

. 這里 0 是單位元素(identity element).

二、環(huán)論（ring)

A ring in the mathmatical sense is a set? together with two binary operations?(分表加法和乘法). 滿足以下條件：

加法結(jié)合性(additive assocativity): for all a, b, c in S, s.t. (a+b)+c = a + (b+c);

加法交換性(additive commutativity): for all a, b in S, s.t. a + b = b + a;

加法單位元(additive identity): there exists an element e in S, s.t. a + e = e + a = a, 這個(gè) e 也可以寫成 0, 加法的單位元是0;

分配性(left and right distributivity): for all a, b, c in S, s.t. a * (b+c) = a*b + a*c,以及 (a + b) *c = a*c + b*c;

加法可逆(additive inverse): for every a in S, there exists -a in S, s.t. a + (-a) = e, e 同上， e = 0.

乘法結(jié)合性(multiplicative associativity)：。滿足該性質(zhì)的環(huán)也叫 associative ring.

乘法交換性(mulitplicative commutativity):?. (a ring satisfying this property is termed a commutative ring).

乘法單位元(multiplicative identity): There exists an element?.(a ring satisfying this property is termed a unit ring, or a ring with identity).

乘法可逆(multiplicative inverse):?. Where 1 is the identity element.

定義一個(gè)環(huán)至少要滿足前五個(gè)條件，但一般會(huì)加上第六個(gè)條件變成associative ring. 沒(méi)有滿足第六個(gè)條件的叫nonassociative ring.

滿足所有條件的就稱為域(field).不滿足 7 乘法交換的環(huán)叫做 division algebra(可除代數(shù))也叫 skew field(非交換域).

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