Space of prime congruences in tropical geometry

Abstract

We investigate spaces of prime congruences on tropical algebras and study their geometry, inspired by classical scheme theory. Our main strategy is to use tropical algebras associated to ordered monoids, which play the role of the monomial structure of these algebras. Using the space of prime congruences as local models, we introduce a tropical toric scheme which contains the usual tropical toric variety as a subspace. We show how the separatedness and properness of these schemes are captured by scheme-theoretic points. As an application of our framework, we obtain a necessary and sufficient condition for a prime congruence to be finitely generated.