Properties

Archimedean

The Archimedean property of the real numbers can be generalized to partially ordered groups.

:Property: A partially ordered group is called Archimedean when for any , if and for all then . Equivalently, when , then for any , there is some such that .

Integrally closed

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.

See also

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Note

References

*M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
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*M. R. Darnel, The Theory of Lattice-Ordered Groups, Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
*L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1963.
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*V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish), Fully Ordered Groups, Halsted Press (John Wiley & Sons), 1974.
*V. M. Kopytov and N. Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, 1996.
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*R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
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