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.9 t2 j6 s6 ]* D5 v( q% 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+. 3 z7 }/ m+ a K$ q* J Q; i , E4 L. h0 s \$ U) D. E* Q; b0 KBy the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. 5 `& r# D( r+ p2 w) V* s- d6 a" ^: m4 b5 `0 D9 U3 G
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: : p- J9 ?4 z, }! U. ? ; v/ M d9 R) t5 s0 ∈ H ( J7 M- K' f% I {if a ∈ H and b ∈ H then a+b ∈ H $ U3 z! w5 W7 U: q7 W+ I zif a ∈ H then -x+a+x ∈ H for each x of G 1 Y( O6 F3 K9 S/ U0 w5 K% \
if a ∈ H and -a ∈ H then a=0 ) _/ U5 `3 l6 }) q; q1 @
Examples ' F% x! R: D- C# K7 i GAn ordered vector space is a partially ordered group # s+ L/ L: b8 r5 D. q8 s! v @
A Riesz space is a lattice-ordered group ' a. x& G! \/ h! P# Q1 P" q. 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. + b* U9 Z1 x$ d6 @: ~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. " q' I: Y9 }$ v4 }3 C, Z序线性空间是有序群6 U* x1 u- {. r