Difference between revisions of "Vector group"
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| − | where this intersection is taken over all combinations of signs | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)</TD></TR></table> | ||
Latest revision as of 08:28, 6 June 2020
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=15917
Vector group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_group&oldid=15917
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