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Preordered groups and valued fields

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arxiv 1912.03928 v1 pith:PIZXC3NT submitted 2019-12-09 math.GR math.ACmath.CO

classification math.GRmath.ACmath.CO
keywords fieldpreordersvaluationsalgebraicanalogiescantorcombinatorialcountable
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abstract

We study algebraic, combinatorial and topological properties of the set of preorders on a group, and the set of valuations on a field. We show strong analogies between these two kinds of sets and develop a dictionary for these ones. Among the results we make a detailed study of the set of preorders on $\mathbb Z^n$. We also prove that the set of valuations on a countable field of transcendence degree at least 2 is an ultrametric Cantor set.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Spaces of left-preorders on free products

    math.GR 2025-07 conditional novelty 6.0 of 10

    Left-preorder spaces of free products have no isolated elements; when the factors are finitely generated, every nonempty such space is a Cantor set.

  2. The space of Conradian left-preorders

    math.GR 2025-06 conditional novelty 6.0 of 10

    For any group, the space of Conradian left-preorders is either finite or uncountable, and finiteness is characterized by the existence of a unique finite rational series without abelian jumps.

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