The spectrum of a localic semiring

TL;DR

This paper introduces a new spectrum construction for localic semirings, generalizing classical spectra from algebraic structures and providing a constructive, quantalic perspective that captures additional differential information.

Contribution

It defines a spectrum for localic semirings using symmetric monoidal category arguments, unifying and extending classical spectrum constructions in a constructive framework.

Findings

Spectrum constructed as frame of overt weakly closed radical ideals

Reduces to classical spectra in special cases

Provides a quantalic spectrum with potential differential information

Abstract

A number of spectrum constructions have been devised to extract topological spaces from algebraic data. Prominent examples include the Zariski spectrum of a commutative ring, the Stone spectrum of a bounded distributive lattice, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann-Lawson spectrum of a continuous frame. Inspired by the examples above, we define a spectrum for localic semirings. We use arguments in the symmetric monoidal category of suplattices to prove that, under conditions satisfied by the aforementioned examples, the spectrum can be constructed as the frame of overt weakly closed radical ideals and that it reduces to the usual constructions in those cases. Our proofs are constructive. Our approach actually gives 'quantalic' spectrum from which the more familiar localic spectrum can then be derived. For a discrete ring this yields the quantale of ideals and in general should contain additional 'differential' information about the semiring.