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lilianjie        

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    2012-1-13 11:05
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    [LV.4]偶尔看看III

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    发表于 2012-1-9 13:53 |只看该作者 |倒序浏览
    |招呼Ta 关注Ta
    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.& G) q6 G' f9 m5 ]$ r
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    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 h: R+ a& N. K3 B2 y( y1 `- k
    By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b." R. G3 g: N6 R. v6 A+ V8 {* M
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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:$ N1 D: e$ i9 d1 S* n

    % N% I) f2 \/ r/ `0 ∈ H
    2 K9 _! B3 K2 v( `$ kif a ∈ H and b ∈ H then a+b ∈ H
    2 b! x7 x  ?& n1 n: I0 o( O; tif a ∈ H then -x+a+x ∈ H for each x of G
    7 X7 _( m; i5 Bif a ∈ H and -a ∈ H then a=0
    / v/ a1 t+ d, B9 K  |6 [3 c  F
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    lilianjie        

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    [LV.4]偶尔看看III

    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:
    6 M  @' _0 g# M0 Z2 d# @+ z$ h& o6 O0 T$ R+ Z0 [: [* {6 F( v
    若 a < 0,则 − a > 0。 3 l' q0 f# z$ H. B
    若 a,b > 0,则 a + b > 0。
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    lilianjie        

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    [LV.4]偶尔看看III

    Examples
    ; A$ y) k( ^% ^  L8 n3 hAn ordered vector space is a partially ordered group + e! _3 t" c# W" V( ]  \
    A Riesz space is a lattice-ordered group 3 l1 R! ^* u0 b" e* K
    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.
    0 O( C. L, U1 X8 w+ \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. ( F; j+ P2 G; H2 V. o; W8 ~: |
    序线性空间是有序群) a+ W) I" [. w. I+ v( ]5 H1 h
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    Z/R/R*都是有序交换群
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