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.

: G: R' X1 |! OAn 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:

! e- c9 j0 |6 A' L  J0 ∈ H
if a ∈ H and b ∈ H then a+b ∈ H
- C1 ^3 Z2 J4 U, ~# jif a ∈ H then -x+a+x ∈ H for each x of G
% l2 _1 m2 i( O8 T7 z: Eif 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：
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3 N0 A- v, p" S/ q- o# ?若 a < 0，则 − a > 0。
6 [" ?3 r  }9 f$ n+ D3 Q8 F若 a,b > 0，则 a + b > 0。

lilianjie

Examples
An ordered vector space is a partially ordered group
+ D9 v; }* }7 o' j/ T3 [6 V& k# @A Riesz space is a lattice-ordered group
6 T- ]8 w# z; M4 h7 t( p* iA 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.
$ M0 R3 X6 n! q' u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