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.
/ S$ y3 f1 t# y% x1 _2 q0 p) q$ {/ W
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+.
9 \+ k! O. ^  j
By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.
2 \8 U) V4 _3 v
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:
% s8 A( I+ x- `! P1 b  R
% H- u9 m; S7 k3 R1 h& I3 u0 ∈ H
if a ∈ H and b ∈ H then a+b ∈ H
if a ∈ H then -x+a+x ∈ H for each x of G
6 Q; |9 p. {0 _" V4 p4 x' O/ Qif a ∈ H and -a ∈ H then a=0

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：

! [$ G% k8 u# B$ C" A. @; y若 a < 0，则 − a > 0。
若 a,b > 0，则 a + b > 0。
; e* V. {$ p# M4 }

lilianjie

Examples
An ordered vector space is a partially ordered group
. `$ l1 J* W0 F' nA Riesz space is a lattice-ordered group
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.
$ s" `8 _- |: N/ ]. b8 I3 ?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
4 U6 I! a9 W! `& g
Z/R/R*都是有序交换群

孤寂冷逍遥