25:
516:
Ehrlich, Philip (1995), "Hahn's "Über die nichtarchimedischen Grössensysteme" and the
Origins of the Modern Theory of Magnitudes and Numbers to Measure Them", in Hintikka, Jaakko (ed.),
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Sitzungsberichte der
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From
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464:"lo.logic - Hahn's Embedding Theorem and the oldest open question in set theory"
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is just the group of real numbers. Then Hahn's
Embedding Theorem reduces to
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566:
Clifford, A.H. (1954), "Note on Hahn's
Theorem on Ordered Abelian Groups",
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490:, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.:
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it is a subgroup of the ordered additive group of the real numbers).
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405:'s theorem (which states that a linearly ordered abelian group is
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The theorem states that every linearly ordered abelian group
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nonzero elements are
Archimedean-equivalent. In this case,
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Hausner, M.; Wendel, J.G. (1952), "Ordered vector spaces",
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is "infinitesimal" with respect to the other. The group
533:(1907), "Über die nichtarchimedischen Größensysteme.",
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Gravett, K. A. H. (1956), "Ordered
Abelian Groups",
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588:Proceedings of the American Mathematical Society
568:Proceedings of the American Mathematical Society
525:, Kluwer Academic Publishers, pp. 165–213
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430:together provide another proof. See also
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394:{\displaystyle \mathbb {R} ^{\Omega }}
221:{\displaystyle \mathbb {R} ^{\Omega }}
147:{\displaystyle \mathbb {R} ^{\Omega }}
16:Description of linearly ordered groups
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548:The Quarterly Journal of Mathematics
486:Fuchs, László; Salce, Luigi (2001),
488:Modules over non-Noetherian domains
89:dealing with ordered structures on
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97:gives a simple description of all
38:it lacks sufficient corresponding
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422:of the theorem. The papers of
99:linearly ordered abelian groups
281:, exactly one of the elements
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492:American Mathematical Society
418:gives a clear statement and
251:{\displaystyle \mathbb {R} }
173:{\displaystyle \mathbb {R} }
85:, especially in the area of
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184:(with its standard order),
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273:. For any nonzero element
293:|. Two nonzero elements
180:is the additive group of
623:Theorems in group theory
432:Fuchs & Salce (2001
258:which vanish outside a
53:more precise citations.
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307:Archimedean equivalent
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101:. It is named after
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340:| > |
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594:: 977–982,
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358:Archimedean
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51:introducing
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473:2021-01-28
450:References
320:such that
285:or −
34:references
554:: 57–63,
541:: 601–655
387:Ω
370:singleton
230:functions
214:Ω
140:Ω
103:Hans Hahn
531:Hahn, H.
438:See also
158:, where
123:subgroup
119:embedded
109:Overview
510:1794715
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523:(PDF)
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