# Vector group

A partially ordered group which is imbeddable in a complete direct product of totally ordered groups (cf. 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 $G$ 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

 [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
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