Partially ordered group

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. A group G is a partially ordered group if and only if there exists a subset H (which is G⁺) of G such that:

A partially ordered group G with positive cone G⁺ is said to be unperforated if n · g ∈ G⁺ for some positive integer n implies g ∈ G⁺. Being unperforated means there is no "gap" in the positive cone G⁺.