Partially-Ordered Group
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.