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Asymptotic Differential Algebra and Model Theory of Transseries$
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Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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Valued Abelian Groups

Valued Abelian Groups

(p.70) Chapter Two Valued Abelian Groups
Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Princeton University Press

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more detail. Ordered abelian groups occur as value groups of valued fields, whereas valued abelian groups arise because the logarithmic derivative map on a valued differential field like induces a valuation on the value group that turns out to be very useful. Furthermore, the notion of a pseudocauchy sequence makes perfect sense in the general setting of valued abelian groups, and the basic facts about these sequences yield a natural proof of a generalized Hahn Embedding Theorem. The chapter also considers valued vector spaces, including spherically complete valued vector spaces, and proves a version of the Hahn Embedding Theorem for valued vector spaces. Special attention is given to particularly well-behaved valued vector spaces known as Hahn spaces.

Keywords:   valued abelian group, ordered set, ordered abelian group, valued field, valued differential field, valuation, pseudocauchy sequence, Hahn Embedding Theorem, valued vector space, Hahn space

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