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* @1 \ W% N5 g9 O$ F% K( j# Q' E% A
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+.# C1 q5 V4 \! x; }; Z" k, P, A* Q7 e
3 q, T8 g6 `5 Z# p" Y0 D1 f
By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.8 U) F/ Z7 \/ Y, F
' O4 f2 X& C: i8 H
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: ! s' ~+ X- h/ i' m' V! H9 H+ b* d1 N8 U1 T
0 ∈ H - o5 ~, m; z a" H; P
if a ∈ H and b ∈ H then a+b ∈ H ( D5 I: q) ^1 N( ^7 @
if a ∈ H then -x+a+x ∈ H for each x of G [: O8 x/ X( d! k+ _* x5 e
if a ∈ H and -a ∈ H then a=0 & |3 v. j; A9 m7 ^1 \& b# g" V9 _. |1 [ 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：" \' e/ L+ ?: b- S3 x9 O
& t0 ]- \& Q7 U9 |+ O0 Z' K' v" q* j- E
若 a < 0，则 − a > 0。 ( Y- Z6 I! r: n( B5 y# s" V$ R3 I9 N
若 a,b > 0，则 a + b > 0。 * l9 v" E( Q' Y' p! l7 j( v$ N 作者: lilianjie 时间: 2012-1-9 13:59
Examples% B9 s' Z: w% I
An ordered vector space is a partially ordered group 6 w! s: @4 o$ c7 w' L
A Riesz space is a lattice-ordered group ) [' D4 K% V! s& k! C: J% |5 t2 iA 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 n# W' X, D% iMore 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. 2 H& k& k, F* O4 L5 r v& h8 a Z序线性空间是有序群 x5 K* Y0 x* d5 E; e. Y! @1 L0 u; f! O" K
Z/R/R*都是有序交换群作者: 孤寂冷逍遥 时间: 2012-1-9 17:48
