High Quality Content by WIKIPEDIA articles! 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. 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+. By the definition, we can reduce the partial order to a monadic property: a ? b if and only if 0 ? -a+b. For the general group G, the existence of a positive cone specifies an order on G.

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