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 .
|||L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)|
Vector group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_group&oldid=15917