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.

1 o; `/ K, a( E  J* i+ l& x/ 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+.

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

" M1 U" |7 `8 {: F# }+ ~* fFor 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:
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5 z+ w! D4 g- t! q8 V. Z0 ∈ H
) i; x8 i& V& A) b4 B& w: Bif a ∈ H and b ∈ H then a+b ∈ H
6 E3 o( C" B* Q$ a& Cif 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：
( ~  V+ p( }8 U* ~
$ k0 K& l6 U* _. T若 a < 0，则 − a > 0。
若 a,b > 0，则 a + b > 0。
3 {6 `: n1 Q) g& x' Q8 a/ u. ^: n

lilianjie

Examples
An ordered vector space is a partially ordered group
A Riesz space is a lattice-ordered group
/ K0 R" d& A* ^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.
8 _; t% f- T* ^8 {1 j: e" A* rMore 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

Z/R/R*都是有序交换群

孤寂冷逍遥