Preordered groups and valued fields

2019/12/09 by Julie Decaup, Decaup, Julie, Guillaume Rond +1 · 2 citations
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1912.03928

openalex publication_date 2019/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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 Zⁿ. We also prove that the set of valuations on a countable field of transcendence degree at least 2 is an ultrametric Cantor set.

Cited by

• Spaces of left-preorders on free products
• The space of Conradian left-preorders

Related

• Some algebraic and topological properties of subspaces of (pre)orders
• The algebraic theory of valued fields
• Definable ranks
• Corps valués locaux
• Equations over valued groups