Some functor calculus on semirings

TL;DR

This paper introduces a functorial framework for ideal theory in commutative semirings, utilizing spectral spaces and coherent frames to unify spectral constructions through categorical and pointfree methods.

Contribution

It develops radical ideal functors and a support-based reconstruction approach, connecting spectral and lattice-theoretic structures in semiring theory.

Findings

Radical ideal functor yields coherent frames.

The $k$-radical ideal functor forms a dense sublocale.

Spectral and lattice-theoretic reconstructions are achieved.

Abstract

We develop a functorial framework for the ideal theory of commutative semirings using coherent frames and spectral spaces. Two central constructions-the radical ideal functor and the $k$-radical ideal functor-are shown to yield coherent frames, with the latter forming a dense sublocale of the former. We define a natural transformation between these functors and analyze their categorical and topological properties. Further, we introduce a notion of support that assigns to each semiring a bounded distributive lattice whose spectrum is homeomorphic to its prime spectrum. This enables the reconstruction of the radical ideal frame via lattice-theoretic data. Applications include adjunctions between quantales and complete idealic semirings; and a comparison of prime and $k$-prime spectra in semisimple $r$-semirings. Our results unify various spectral constructions in semiring theory through a categorical and pointfree perspective.