标题: 有序群/有序交换群 [打印本页] 作者: lilianjie 时间: 2012-1-9 13:53 标题: 有序群/有序交换群 In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.& R! p: [# b9 q' V' f: G. Q8 T
3 {. @# E2 }5 u! W1 k/ Z$ [& Y* I' A$ j
An element x of G is called positive element if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and it is called the positive cone of G. So we have a ≤ b if and only if -a+b ∈ G+.+ Q: @5 T1 x+ Y$ P" n; \
4 H* X5 H2 Y( n4 |" Y3 }& U% S
By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. - u' t: q1 G0 `! c- f* @( ?% ~( A) B( h+ k
For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially ordered group if and only if there exists a subset H (which is G+) of G such that:4 f! I4 v2 J0 \2 I1 Z
+ q, a* h! J- l8 g* \
0 ∈ H * U: q" P) e0 I( I6 Pif a ∈ H and b ∈ H then a+b ∈ H 6 v5 ]8 d- g3 _6 c
if a ∈ H then -x+a+x ∈ H for each x of G + F) N7 E$ m; S5 N5 e& y" z
if a ∈ H and -a ∈ H then a=0 3 _( H* W8 x/ G# D; ~ 作者: lilianjie 时间: 2012-1-9 13:53
有序交换群系指一对 (Γ, > )，其中 Γ 为交换群， > 为其上的一个二元关系，且满足如下条件：- A* v/ l$ x1 A9 K5 B
9 H+ v' s( q( U) L' l9 v/ j& P+ _/ v
若 a < 0，则 − a > 0。 & p2 R0 I1 Z! m
若 a,b > 0，则 a + b > 0。 1 D4 }) N; d. z- D& b 作者: lilianjie 时间: 2012-1-9 13:59
Examples 2 J0 n" P! U4 s; u5 b+ s- q1 i6 j# wAn ordered vector space is a partially ordered group ; b/ V/ o! q7 ?1 t; h7 F2 ]
A Riesz space is a lattice-ordered group 7 w' q+ f) y# |5 ]' d( W; V
A typical example of a partially ordered group is Zn, where the group operation is componentwise addition, and we write (a1,...,an) ≤ (b1,...,bn) if and only if ai ≤ bi (in the usual order of integers) for all i=1,...,n. / B0 L! H/ D! U: N! l6 U* gMore generally, if G is a partially ordered group and X is some set, then the set of all functions from X to G is again a partially ordered group: all operations are performed componentwise. Furthermore, every subgroup of G is a partially ordered group: it inherits the order from G. $ y3 h" k6 i2 z8 Y, ^. s' [/ W序线性空间是有序群6 A% {/ k3 K+ ]6 ^
+ O X: ]" ]/ @/ y: [
Z/R/R*都是有序交换群作者: 孤寂冷逍遥 时间: 2012-1-9 17:48
