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. 0 m# ]7 Q V9 G- g8 L5 p7 O# w j% N# R& q# |0 i! p6 EAn 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+. 1 c9 } k6 I; i; p. b$ ^# W9 @, h" { 7 D8 n1 k1 V) q% F( cBy the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. 2 x7 z: x3 l4 D- X ! a1 c3 L' P' K" U, z0 _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: / b4 v W2 A; l* T( D' _: Z ) j. s! m. }) V- m$ _' z1 w* L' N0 ∈ H & X3 z* h% Y5 }! ^) _if a ∈ H and b ∈ H then a+b ∈ H 8 T/ h/ g5 f" e; ?) A& S, r7 Yif a ∈ H then -x+a+x ∈ H for each x of G ) m p% q3 k3 f6 T4 \if a ∈ H and -a ∈ H then a=0 ; v# s7 r& N6 k 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：# K5 j* Q5 j) a7 D
% k/ I" t4 ^% }若 a < 0，则 − a > 0。 4 t5 U; w+ x. A若 a,b > 0，则 a + b > 0。 ; T; L6 x, K8 D/ f! x+ e* {: k* ]作者: lilianjie 时间: 2012-1-9 13:59
Examples * T( ?$ r% D- G& QAn ordered vector space is a partially ordered group 4 S+ |, B% K; jA Riesz space is a lattice-ordered group % g0 A6 k) }5 eA 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. + E& z# z# M+ B& l4 q1 B ]! Z
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. ( z) H! c+ C! B2 z* L% ^
序线性空间是有序群 0 Z% E2 @" n' ~1 a- G7 q% A( K" n% P/ w( [
Z/R/R*都是有序交换群作者: 孤寂冷逍遥 时间: 2012-1-9 17:48
