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.1 e( j; ~, N2 J3 V
# M& @( t4 p/ t
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+. ' `2 C" O" s% {, |4 i- b9 Y& K5 P2 d: k, o
By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.4 [+ ~) R& O! ^8 B5 `* j; p
5 d$ @% q" ]5 g3 ]6 hFor 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:! [) V, E. p3 @, a
0 M4 E+ V& _, y8 \9 s1 R
0 ∈ H + E3 \; N5 O2 l+ M1 d! B) iif a ∈ H and b ∈ H then a+b ∈ H 4 [' Z% ~ ]" Z# O e
if a ∈ H then -x+a+x ∈ H for each x of G 1 h/ M+ w+ f# R
if a ∈ H and -a ∈ H then a=0 : `/ I+ m0 L7 H& {, u+ z' e7 C5 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：# X% F3 r q3 |/ m
4 l7 S4 }' Q/ W# { I若 a < 0，则 − a > 0。 1 m* @2 f+ n5 W5 Y* w
若 a,b > 0，则 a + b > 0。 - a* E- J, B9 S% [) a0 \作者: lilianjie 时间: 2012-1-9 13:59
Examples ' C+ w5 F0 \* X/ ^7 m' o0 QAn ordered vector space is a partially ordered group % L/ B( z, `8 J8 P/ @
A Riesz space is a lattice-ordered group 8 k9 e5 _1 B6 D# p5 @: s6 R
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 U, [: H8 ]6 {- l3 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. ( S& d1 ?, @% b; X
序线性空间是有序群8 j2 y% I& y- w8 Y
