Vector group

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A partially ordered group which is imbeddable in a complete direct product of totally ordered groups (cf. Totally ordered group). A group is a vector group if and only if its partial order is an intersection of total orders on . A partially ordered group will be a vector group if and only if its semi-group of positive elements satisfies the following condition: For any finite collection of elements of ,

where this intersection is taken over all combinations of signs , while denotes the smallest invariant sub-semi-group of containing . An orderable group is a vector group if and only if for any it follows from that .


[1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Vector group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. KokorinV.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article