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. " H9 m! h7 b- |5 v$ Q 1 W, C. H: V* X S0 T& p1 {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+.. F& h- ] k) {9 Z& y( [& e* 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.: i) k) k9 y5 q4 B0 E4 F
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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: $ D* s. b6 \% t+ t- h0 S : |$ _) T O# I. D( I. r" H3 e0 ∈ H 5 l# P, K& s: O7 N- o7 ?# v4 D
if a ∈ H and b ∈ H then a+b ∈ H 9 _; ?" C2 Y2 K4 j/ D3 o
if a ∈ H then -x+a+x ∈ H for each x of G ) Q4 r% ` Q) z% }" B& ^6 \6 A7 Iif a ∈ H and -a ∈ H then a=0 7 m! v/ C$ C2 E# ]& U8 O* |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： , l+ Q; a' X6 }& q- W& ^# m0 a; B8 W& z! u7 M
若 a < 0，则 − a > 0。 & y( v/ r Z; R, O3 t若 a,b > 0，则 a + b > 0。 . A: C8 q. W2 l作者: lilianjie 时间: 2012-1-9 13:59
Examples 6 Q4 h( E% j$ a# w* a% FAn ordered vector space is a partially ordered group - X2 _" M0 Y0 a* iA Riesz space is a lattice-ordered group 4 y# n* x( I5 T0 X4 V& @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. 9 ^6 u+ \ [% L; e9 UMore 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. % v- E) M# L: b4 N/ p2 v& [
序线性空间是有序群7 ~- l% y( |: T0 ~' n
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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
