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In abstract algebra, a partially ordered group is a group 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 if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G, and is called the positive cone of G.
By translation invariance, we have a ≤ b if and only if 0 ≤ -a + b.So we can reduce the partial order to a monadic property: a ≤ b if and only if -a + b ∈ G.
For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially orderable group if and only if there exists a subset H of G such that:
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