Vector group
From Encyclopedia of Mathematics
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
,
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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
.
References
[1] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |
How to Cite This Entry:
Vector group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_group&oldid=15917
Vector group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_group&oldid=15917
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