ordered group
Definition 1. We say that the subsemigroup of the group (with the operation denoted multiplicatively) defines an , if
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where and the members of the union are pairwise disjoint.
The order “” of the group is explicitly given by setting in :
Then we speak of the ordered group , or simply .
Theorem 1.
The order “” defined by the subsemigroup of the group has the following properties.
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1.
For all , exactly one of the conditions holds.
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2.
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Definition 2. The set is an ordered group equipped with zero 0, if the set of its elements distinct from its element 0 forms an ordered group and if
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Cf. 7 in examples of semigroups.
References
- 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
- 2 Paul Jaffard: Les systèmes d’idéaux. Dunod, Paris (1960).
Title | ordered group |
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Canonical name | OrderedGroup |
Date of creation | 2013-03-22 14:54:36 |
Last modified on | 2013-03-22 14:54:36 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 16 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 06A05 |
Classification | msc 20F60 |
Related topic | KrullValuation |
Related topic | PartiallyOrderedGroup |
Related topic | PraeclarumTheorema |
Defines | ordered group equipped with zero |