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. ?/ V, `% I% E$ ~. o# L/ ?: T* n. h, t! _- [
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+. 6 T7 N' E7 S' n# j: h" h! } ; S) y, h( o* n3 W4 L7 kBy the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. # b7 t1 R0 j D$ u, n& P7 q0 Q2 f1 Y o$ x
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:2 K0 d/ W! L: }) u( P2 Z
( a7 T" B j9 E: f c8 X) S
0 ∈ H 3 @2 w8 \1 ]4 v/ P0 g% u1 V
if a ∈ H and b ∈ H then a+b ∈ H 8 S( S A' I5 p( k a; l( p" S
if a ∈ H then -x+a+x ∈ H for each x of G 0 _5 V8 ?, Y- I' g
if a ∈ H and -a ∈ H then a=0 ' t# N5 M7 { C7 r; {
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An ordered vector space is a partially ordered group ( t* v7 ^: @, i* _
A Riesz space is a lattice-ordered group : L$ q: g2 l6 `2 S
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. ; z+ E1 N1 i, _+ `8 d/ N& w7 IMore 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 s2 g& E2 E, a* B) R
序线性空间是有序群 $ i8 J: U6 Y* x) [) [7 @/ U4 H& _9 o! m0 e( R- N, b! u9 z7 q) ~) E
Z/R/R*都是有序交换群