Congruence spectra and geometric reconstruction of $\mathbf{T}$-algebras

Abstract

We develop a geometric theory of congruence spectra for semirings. We define the congruence spectrum for a semiring as the set of prime congruences on it, equip it with the Zariski topology, and construct a structure sheaf. For a class of semirings including $\mathbf{B}$-algebras, we establish quasi-compactness of their congruence spectra. We then prove reconstruction results for congruence spectra of quotients $K/\mathbf{E}(V)$, where $V$ is a subset of Euclidean space, $K$ is a tropical Laurent polynomial semiring, a tropical Laurent polynomial function semiring, or a tropical rational function semifield, and $\mathbf{E}(V)$ is the congruence of pairs of elements of $K$ that agree on $V$. The global sections of the structure sheaf recover $K/\mathbf{E}(V)$, and the corresponding algebraic and geometric categories are contravariantly equivalent. For finite unions of $\mathbf{R}$-rational polyhedral sets $V$, we completely classify the prime congruences on $K/\mathbf{E}(V)$. As an application, we study rational function semifields of tropical curves with parallel rays. We show that their congruence spectra recover the underlying curves together with the parallelism of rays and their intrinsic metrics.

arXiv comment17 pages, one of the main corrections in v2 is Example 3.1; the main results are unchanged