# Integrally closed ordered group

In algebra, an ordered group *G* is called **integrally closed** if for all elements *a* and *b* of *G*, if *a*^{n} ≤ *b* for all natural *n* then *a* ≤ 1.

This property is somewhat stronger than the fact that an ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent. There is a theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed.

## References

- A. M. W. Glass,
*Partially Ordered Groups*, World Scientific, 1999

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