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.

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

0 ∈ H
$ r9 k& c( J: {9 V  P* Z  s: b9 Pif a ∈ H and b ∈ H then a+b ∈ H
+ d, ^: T9 b/ T3 lif a ∈ H then -x+a+x ∈ H for each x of G
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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  U  J5 @, P: B  |8 ^! l若 a < 0，则 − a > 0。
5 Y/ P' C" J# @- e0 v) A若 a,b > 0，则 a + b > 0。

lilianjie

Examples
An ordered vector space is a partially ordered group
  t1 c; _# ^: j2 c9 KA 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.
: t3 F; G8 _; ^1 F# s/ ]% K- zMore 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
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Z/R/R*都是有序交换群

孤寂冷逍遥