Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

TL;DR

This paper characterizes spectral spaces via prime spectra of idempotent semirings, establishing a categorical equivalence, and explores closure operations in the algebraic structure of semirings, with applications to tropical geometry.

Contribution

It extends Hochster's theorem to idempotent semirings, constructs a categorical equivalence with spectral spaces, and introduces new closure operations for semirings.

Findings

Spectral spaces are exactly the prime spectra of idempotent semirings.

The space of valuations and prime congruences on an idempotent semiring are spectral.

New closure operations, including integral and Frobenius closures, are developed for semirings.

Abstract

Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the \emph{prime $k$-spectrum} of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the \emph{space of valuations} and the \emph{space of prime congruences} on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of \emph{closure operations} for semirings, and provide several examples. In particular, we introduce an \emph{integral closure operation} and a \emph{Frobenius closure operation} for idempotent semirings.