On the Tree Structure of Orderings and Valuations on Rings

TL;DR

This paper introduces a unified framework for quasi-orderings on rings, showing they form a rooted tree structure and are topologically spectral, unifying orderings and valuations in a broad algebraic context.

Contribution

It defines a new notion of quasi-orderings on rings, establishes their tree structure, and links this to spectral space properties, unifying orderings and valuations.

Findings

The set of all quasi-orderings forms a rooted tree under a modified coarsening relation.

The space of quasi-orderings is spectral, isomorphic to the spectrum of a commutative ring.

The paper studies the topological properties of the quasi-orderings space.

Abstract

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 $\leq$ on the set $\mathcal{Q}(R)$ of all quasi-orderings on $R.$ One of our main results states that $(\mathcal{Q}(R),\leq')$ is a rooted tree for some slight modification $\leq'$ of $\leq,$ 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 $(\mathcal{Q}(R),\leq')$ is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with $1.$ We conclude this paper by studying $\mathcal{Q}(R)$ as a topological space.