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.
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) r  v2 s$ S2 K! X; e8 a( JAn 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+.

/ w( `& X4 }( [! q9 ?7 X/ iBy 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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& _0 `& j0 u+ gFor 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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if a ∈ H and b ∈ H then a+b ∈ H
" d8 y! Z0 N- y' p, Aif a ∈ H then -x+a+x ∈ H for each x of G
5 A' M; s$ |* b/ y) Pif 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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若 a < 0，则 − a > 0。
若 a,b > 0，则 a + b > 0。
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lilianjie

Examples
9 q$ T8 z, h) B; ]. }. R9 W3 A$ }An ordered vector space is a partially ordered group
, G3 O4 }( `' R) T0 P& u9 \A Riesz space is a lattice-ordered group
8 \# v5 a6 l9 k; h  V0 bA 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.
& ]7 k4 ^' e6 Q; t3 }' ?7 @# K7 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

Z/R/R*都是有序交换群

孤寂冷逍遥