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)