标题: 有序群/有序交换群 [打印本页] 作者: lilianjie 时间: 2012-1-9 13:53 标题: 有序群/有序交换群 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.! n# K0 G& F, N8 {
0 B3 `" R% Z* f% r- UAn 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+. / ?6 G' T, u2 T( c" m + W4 ]8 ?0 n+ X* z% l: @By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. & _& K8 G2 {$ @ F! M % R# ?- |: ?6 dFor 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: % |" T) E7 R3 m& a! `) H7 h6 Y: c) N, c) E, B9 Y- I' r% i
0 ∈ H - X ^3 F" K- s" {if a ∈ H and b ∈ H then a+b ∈ H : E r1 l: M) v0 |6 O
if a ∈ H then -x+a+x ∈ H for each x of G $ E, L* V2 @: h# h3 O8 c4 qif a ∈ H and -a ∈ H then a=0 $ L+ S# N9 \( h作者: lilianjie 时间: 2012-1-9 13:53
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：9 ?3 C- S( J/ Y# M3 Z
3 t* F& ^+ L1 }4 z( P7 t; c若 a < 0，则 − a > 0。 s+ i! P! k f% X
若 a,b > 0，则 a + b > 0。 8 W* X+ Z1 o; s7 N作者: lilianjie 时间: 2012-1-9 13:59
Examples : i$ ~% R( h, E+ GAn ordered vector space is a partially ordered group * \/ L1 H9 O7 I; D9 o8 dA Riesz space is a lattice-ordered group ! _% l$ _# g/ l1 a' ]4 ?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. & v$ d. J6 ?# V
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. 4 h' Z: Q0 k4 n2 O4 w, s序线性空间是有序群# ]3 n3 V: E" k( [' m
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Z/R/R*都是有序交换群作者: 孤寂冷逍遥 时间: 2012-1-9 17:48
