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

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. This is work in progress with Cristhian Garay Lopez.