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. 1 y6 u) \) w" n. t2 Y- g y& R- q" I, j$ ^3 s0 Q l
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+.) x& w1 W2 Q# P/ y: s$ y8 Y
0 J) a$ J# |$ q: @" u. }By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.2 p, V6 q) q$ b+ f+ O
% Z& [! }8 {- h$ r: T+ ?9 S rFor 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: $ V* [' G2 R) t) e; g$ Z5 Z5 k8 t9 e7 {* |
0 ∈ H : H5 R9 H1 O% t% `+ Y- n5 b* o8 R* wif a ∈ H and b ∈ H then a+b ∈ H $ [5 B$ |5 ^$ V. H
if a ∈ H then -x+a+x ∈ H for each x of G 2 S# o3 }2 z* o* X, M. n# k4 h
if a ∈ H and -a ∈ H then a=0 ! G% a- I9 A; V8 E' Q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：3 r. j* ~" H! }4 F
, g6 g/ c' k* Q若 a < 0，则 − a > 0。 ' }6 N& @8 `4 e! S若 a,b > 0，则 a + b > 0。 * _5 V; X4 L9 V" v4 L: Y/ M 作者: lilianjie 时间: 2012-1-9 13:59
Examples$ z2 }, T- R; b! U
An ordered vector space is a partially ordered group 0 k5 @( d0 f. t8 u7 _4 {3 i/ [
A Riesz space is a lattice-ordered group " V4 c2 m: {' E% C4 U9 R
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. 4 A+ c$ X3 B+ Q5 bMore 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. ; N" y5 V& k" e4 M& D# F
序线性空间是有序群) C* [' u2 d9 h! ?0 H: w2 w
