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Vector group

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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