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lilianjie        

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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 |正序浏览
    |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.
    6 y- t' ^' Z5 h/ `9 L- [& h
    1 N2 i, g+ K7 BAn 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+.! ?" s* y1 a; C. B4 b
    ! k; Z( ~! ]" F- u
    By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.
    0 \2 m( m  c" J: m: L% @% j8 V5 t3 V
    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:4 \5 @/ k0 b" Y9 q* D$ ]' K
    % p# \3 I2 D' J$ Y
    0 ∈ H / E) W" n9 Z& E$ Q! g1 i; u
    if a ∈ H and b ∈ H then a+b ∈ H
    0 U5 v8 H4 x; Eif a ∈ H then -x+a+x ∈ H for each x of G , F* B/ M3 N# ~" J) n
    if a ∈ H and -a ∈ H then a=0 $ t6 n' h0 {& a4 N- f" E/ F
    zan
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    lilianjie        

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    Examples; K; e5 H( _$ y' G! f
    An ordered vector space is a partially ordered group
    4 k+ W/ D. t" EA Riesz space is a lattice-ordered group
    * K# p2 w% d$ j2 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. % G9 ?) h/ H0 f0 m0 U
    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. 6 T" ]4 [% D2 N. |, z' N
    序线性空间是有序群! }, k, K+ y' e' x

    4 ?  s+ a; H. B4 n, KZ/R/R*都是有序交换群
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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:4 }4 x2 N! S8 t7 Y$ @  J$ _

    ! Q5 h2 ?' C- u' O+ U$ C* h4 ]& H若 a < 0,则 − a > 0。
    + I( G& h) _  f若 a,b > 0,则 a + b > 0。
    * D  t4 D: D  c
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