Ordgruppe

Ordgruppe is a term used in mathematics to denote an ordered group, typically described in English as an ordered group. It refers to a group G equipped with a total order ≤ that is compatible with the group operation. In additive notation one writes (G, +, ≤). The compatibility condition means that for all a, b, c in G, if a ≤ b then a + c ≤ b + c. Equivalently, the set of positive elements P = {g ∈ G : e ≤ g} is closed under addition, and every element lies either in P or is the inverse of an element in P (G = P ∪ −P, with P ∩ (−P) = {e}).

Key properties include the fact that any nontrivial ordered group is torsion-free: if g > e, then ng > e for all positive integers n, so g cannot have finite order. The standard objects are linearly ordered groups, sometimes called totally ordered groups, where the order is total. A common refinement is to consider lattice-ordered groups, where any two elements have a meet and a join.

There are standard examples and non-examples. Examples include the integers (Z, +) with the usual order, and the real numbers (R, +) with the usual order. Finite groups cannot admit a translation-invariant total order other than the trivial one, so nontrivial finite groups are not ordered groups in this sense.

In practice, the term ordered group is widely used, especially in algebra and analysis. Ordered abelian groups are particularly common, since commutativity guarantees that left and right translation invariance coincide. Ordered groups play a foundational role in valuation theory, real algebra, and related areas, where they often appear as value groups or as ambient structures for monotone comparisons. In German texts, the term geordnete Gruppe or Ordnungsgruppe is used, with ordgruppe occasionally appearing as a shorthand.