Stefano Mereta: The space of valuated (pre)orders as the spectrum of a ring

Abstract

After recalling results of Robbiano and Mincheva-Jóo on valuated (pre)orders and prime congruences over some polynomial semirings, we will recall the notion of Zariski-Riemann space of an ordered group. We will then introduce a valuation with target a polyhedral semiring whose unit ball is a finite dimensional, non-noetherian Bèzout domain. We show that the aformentioned Zariski-Riemann space is in bijection with the spectrum of this ring. We will then recall some basics about spectral spaces and frame this result with respect to Hochster theorem and recent work by Jun-Ray-Tolliver.