有序群/有序交换群

lilianjie

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.
6 j6 F0 R. Y0 [
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+.

By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.

+ |6 J$ I& `6 ~8 DFor 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:
( |0 t6 h2 k. w2 \6 Q% x
3 O( l6 n0 J+ Q8 C0 ∈ H
if a ∈ H and b ∈ H then a+b ∈ H
+ b+ Z# {  V+ h6 z: p. wif a ∈ H then -x+a+x ∈ H for each x of G
if a ∈ H and -a ∈ H then a=0
: J* `  J. g2 a0 q/ ^

lilianjie

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：

, b. m! R6 B: V  }8 J1 U若 a < 0，则 − a > 0。
  G- ?, U& q: h9 H5 R若 a,b > 0，则 a + b > 0。

lilianjie

Examples
/ J0 M4 E( S+ r4 J5 D3 ~An ordered vector space is a partially ordered group
" c" A6 j* v; C1 IA Riesz space is a lattice-ordered group
/ l: o* }9 H0 l: NA 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.
More 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.
序线性空间是有序群

1 I% G7 T3 o, B9 A: IZ/R/R*都是有序交换群

孤寂冷逍遥