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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.* u+ }& a! E! W" @  z- ~/ i( t

    1 o; `/ K, a( E  J* i+ l& x/ OAn 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+.# {) d, p; C- R! b$ t7 ]" k  q; [
    5 ?: x# E; R' B' x$ {# o
    By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.  d2 \- s8 g; \8 l

    " M1 U" |7 `8 {: F# }+ ~* fFor 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:
    ; y3 m; K; F! ^) @5 m) q
    5 z+ w! D4 g- t! q8 V. Z0 ∈ H
    ) i; x8 i& V& A) b4 B& w: Bif a ∈ H and b ∈ H then a+b ∈ H
    6 E3 o( C" B* Q$ a& Cif a ∈ H then -x+a+x ∈ H for each x of G 3 K3 s% x+ Z) c- P. x9 j& Z
    if a ∈ H and -a ∈ H then a=0
    ' g3 C5 S( {7 ^. b( H- W/ V
    zan
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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:
    ( ~  V+ p( }8 U* ~
    $ k0 K& l6 U* _. T若 a < 0,则 − a > 0。 3 o6 F: [( A  t6 ]$ |
    若 a,b > 0,则 a + b > 0。
    3 {6 `: n1 Q) g& x' Q8 a/ u. ^: n
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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

    Examples$ t& w7 _" }& q) Y9 Q
    An ordered vector space is a partially ordered group , Z5 s3 l* ?; V( q% R' J. s' N' \
    A Riesz space is a lattice-ordered group
    / K0 R" d& A* ^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 _; t% f- T* ^8 {1 j: e" A* rMore 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. : N9 S; ^5 s0 V) ]  k( {$ o9 F
    序线性空间是有序群8 {. A4 t. o+ P
    : S' t; G1 }0 ?1 x) S) P
    Z/R/R*都是有序交换群
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