A **partially ordered group** (POG) is an algebraic structure that combines the concepts of a group and a partial order. Formally, a group \( G \) is equipped with a binary operation (usually denoted as multiplication or addition) and satisfies the group properties—closure, associativity, existence of an identity element, and existence of inverses.
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A group with a partial order preserved by multiplication on both sides: implies for all . In additive notation this says . Preserving only one side is a different orderability condition.