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.

) n8 [: _' q# d) x7 h0 XAn 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+.
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; U, R+ b  e5 M( g! nBy the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.

- w8 `. V+ N. f6 U; 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:

! F0 I! D- e' Z9 ^/ |0 ∈ H
if a ∈ H and b ∈ H then a+b ∈ H
! M# M; [( ~- S+ aif a ∈ H then -x+a+x ∈ H for each x of G
/ \1 x) D/ B7 \& J( `if a ∈ H and -a ∈ H then a=0
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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：
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; |% g& J6 H' i1 F( \* Z* ^) U若 a < 0，则 − a > 0。
1 y5 [* \0 P$ |$ o% b若 a,b > 0，则 a + b > 0。
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lilianjie

Examples
+ L: A8 D& b% S6 fAn ordered vector space is a partially ordered group
A Riesz space is a lattice-ordered group
6 W9 L; \; j% B1 z. fA 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.
$ a: |, ]! l$ B8 @. KMore 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.
0 _( I+ k: I+ u9 N) d4 J: G: o, [序线性空间是有序群
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6 F2 g, m7 i& RZ/R/R*都是有序交换群

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