Some functor calculus on semirings
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 -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 -prime spectra in semisimple -semirings. Our results unify various spectral constructions in semiring theory through a categorical and pointfree perspective.
Cite
@article{arxiv.2506.13378,
title = {Some functor calculus on semirings},
author = {Pronay Biswas and Amartya Goswami and Sujit Kumar Sardar},
journal= {arXiv preprint arXiv:2506.13378},
year = {2025}
}