Chapter

Valued Abelian Groups

Matthias Aschenbrenner, Lou van den Dries and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published by Princeton University Press

Published in print June 2017 | ISBN: 9780691175423
Published online October 2017 | e-ISBN: 9781400885411
Valued Abelian Groups

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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

Chapter.  24508 words. 

Subjects: Optimisation

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