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.5 a1 p8 d* T, R6 K
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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+., @0 f% {/ }: G% U( Y0 z
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By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. 6 F# |1 ~% g0 _' Q' ]) ] / D1 q. ?; |9 AFor 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: * k" K% U" X3 L" R+ I1 `! N2 Q) A0 f+ n3 s: W
0 ∈ H 9 ^! z! ^& C8 p* r7 n6 Z3 O' q) v! o3 b
if a ∈ H and b ∈ H then a+b ∈ H : D* p$ N2 ~' u1 @/ _
if a ∈ H then -x+a+x ∈ H for each x of G * ]+ f* Q0 s$ ]; B& bif a ∈ H and -a ∈ H then a=0 ! D" ]6 `" c. `! C* F4 O- {
Examples 9 e: U: a3 b" a3 \An ordered vector space is a partially ordered group ; `8 [+ o6 K+ J4 ~3 k: ~A Riesz space is a lattice-ordered group # m4 p, K' e2 B0 k9 F8 C5 } [( y2 ]' \7 |2 sA 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. . m m& ~4 G3 z9 x
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. 9 M( F$ p+ p) P: X
序线性空间是有序群 + [, L( u& a1 }$ Q5 A, d& p f3 _6 P0 @' G2 u9 e) U- d% k. ?4 kZ/R/R*都是有序交换群