Integrally closed ordered group

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

This property is somewhat stronger than the fact that a partially 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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