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 
,
 |
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=49140
Vector group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_group&oldid=49140
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