On the tree structure of orderings and valuations on rings

Simon Müller Fachbereich Mathematik und Statistik, Universität Konstanz, Germany

TBD mathscidoc:2203.43019

Arkiv for Matematik, 59, (1), 165-194, 2021.5
Let R be a not necessarily commutative ring with 1. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on R. We proceed by uniformly defining a coarsening relation ≤ on the set Q(R) of all quasi-orderings on R. One of our main results states that (Q(R),≤′) is a rooted tree for some slight modification ≤′ of ≤, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that (Q(R),≤′) is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with 1. We conclude this paper by studying Q(R) as a topological space.
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  title={On the tree structure of orderings and valuations on rings},
  author={Simon Müller},
  booktitle={Arkiv for Matematik},
Simon Müller. On the tree structure of orderings and valuations on rings. 2021. Vol. 59. In Arkiv for Matematik. pp.165-194. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311104346079457956.
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