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. 6 u! C* q0 ]5 _' [6 v9 X& I' [5 }6 [# G) S
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+. : ? e; B, b& Y, Z ( G$ w# P4 d* Q% x; M& g1 V* GBy the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.: }9 V4 B9 h6 D7 J, Q6 y1 b
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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:- |+ j" G' _1 u* N/ t
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0 ∈ H 1 ]! i5 S+ p( L) a! \3 ? w3 \# J
if a ∈ H and b ∈ H then a+b ∈ H 0 c6 Y# [- f8 m1 b' t: Q; u) m
if a ∈ H then -x+a+x ∈ H for each x of G 6 e9 y9 n% f2 Z! ?if a ∈ H and -a ∈ H then a=0 ! d6 E9 ~9 }& p+ W( Q 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：" S0 O* j) n8 j, M0 c" d
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若 a < 0，则 − a > 0。 5 o' _* j2 o, m( L/ G, r. [
若 a,b > 0，则 a + b > 0。 1 h# g0 J6 O8 J8 B7 Q, R, l8 {/ b 作者: lilianjie 时间: 2012-1-9 13:59
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An ordered vector space is a partially ordered group " u& g3 ^( i2 V7 d
A Riesz space is a lattice-ordered group 3 N# j: ? n& B1 GA 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. - T) U% _: e2 v. t/ B4 M& jMore 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. 5 W: w0 K; h/ n% Z% R+ _ J: @" i序线性空间是有序群 $ c7 D2 Y- ~+ x4 t9 |/ l% a2 H / t- d' m) X1 z8 o. j) IZ/R/R*都是有序交换群作者: 孤寂冷逍遥 时间: 2012-1-9 17:48
