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The spectrum of a localic semiring

Part of: Ordered structures Boolean algebras (Boolean rings) Distributive lattices Connections with other structures, applications General commutative ring theory Basic constructions Topological rings and modules Proof theory and constructive mathematics

Published online by Cambridge University Press:  28 February 2022

GRAHAM MANUELL
GRAHAM MANUELL*
Affiliation:
Centro de Matemática, Universidade de Coimbra, 3001-501, Coimbra, Portugal Email address: graham@manuell.me
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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.

MSC classification

Primary: 54B35: Spectra
Secondary: 06D22: Frames, locales 54H13: Topological fields, rings, etc. 13J99: None of the above, but in this section 54B30: Categorical methods 13A15: Ideals; multiplicative ideal theory 06F07: Quantales 03F65: Other constructive mathematics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 3 , November 2022 , pp. 647 - 668
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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