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A [[Partially ordered group|partially ordered group]] which is imbeddable in a complete direct product of totally ordered groups (cf. [[Totally ordered group|Totally ordered group]]). A group  $  G $
is a vector group if and only if its partial order is an intersection of total orders on  $  G $. 
A partially ordered group will be a vector group if and only if its semi-group  $  P $
of positive elements satisfies the following condition: For any finite collection of elements  $  a _ {1} \dots a _ {n} $
of  $  G $,

$$ 
\cap PS ( a _ {1} ^ {\epsilon _ {1} } \dots
a _ {n} ^ {\epsilon _ {n} } , e )  =  P,
$$

where this intersection is taken over all combinations of signs  $  \epsilon _ {i} = \pm  1 $, 
while  $  S ( x \dots z ) $
denotes the smallest invariant sub-semi-group of  $  G $
containing  $  x \dots z $. 
An [[Orderable group|orderable group]]  $  G $
is a vector group if and only if for any  $  g, g _ {1} \dots g _ {n} \in G $
it follows from  $  gg _ {1}  ^ {-} 1 gg _ {1} \dots g _ {n}  ^ {-} 1 gg _ {n} \in P $
that  $  g \in P $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>