archimedean group in nLab

Contents

Contents

Definition

An archimedean group is a strictly ordered group which satisfies the Archimedean property, in which every positive element is bounded above by a natural number.

So an archimedean group has no infinite elements (and thus no non-zero infinitesimal elements).

Properties

• Every archimedean group is an abelian group and has no bounded cyclic subgroups. Every archimedean group admits an embedding into the group of real numbers.

• Every archimedean group is a flat module and a torsion-free group.

• Every Dedekind complete archimedean group is isomorphic to the integers, if the group is not dense, or the Dedekind real numbers, if the group is dense.

Examples

Archimedean groups include

• integers

• half integers?

• rational numbers

• dyadic rationals

• decimal rationals

• real numbers

Non-archimedean groups include

• p-adic integers

• p-adic numbers

See also

• archimedean property

• archimedean protoring

External links

• Wikipedia, Archimedean group