Dedekind cut
Every Archimedean group has the property that, for every Dedekind cut of the group, and every group element ε &gt; 0, there exists another group element "x" with "x" on the lower side of the cut and "x" + ε on the upper side of the cut. However, there exist non-Archimedean ordered groups with the same property. The fact that Archimedean groups are abelian can be generalized: every ordered group with this property is abelian.