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The spectrum of a localic semiring

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

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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References

Banaschewski, B. and Niefield, S. B.. Projective and supercoherent frames. J. Pure Appl. Algebra 70(1-2) (1991), 45–51.Google Scholar
Bunge, M. and Funk, J.. Constructive theory of the lower power locale. Math. Structures Comput. Sci. 6(1) (1996), 69–83.CrossRefGoogle Scholar
Fawcett, B. and Wood, R. J.. Constructive complete distributivity I. Math. Proc. Camb. Phil. Soc. 107(1) (1990), 81–89.CrossRefGoogle Scholar
Gelfand, I.. Normierte Ringe. Mat. Sb. 9(1) (1941), 3–24.Google Scholar
Henry, S.. Constructive Gelfand duality for non-unital commutative C*-algebras (2015). arXiv preprint, arXiv:1412.2009.Google Scholar
Henry, S.. Localic metric spaces and the localic Gelfand duality. Adv. Math. 294 (2016), 634–688.CrossRefGoogle Scholar
Hofmann, K. H. and Lawson, J. D.. The spectral theory of distributive continuous lattices. Trans. Amer. Math. Soc. 246 (1978), 285–310.CrossRefGoogle Scholar
Jacobson, N.. A topology for the set of primitive ideals in an arbitrary ring. Proc. Natl. Acad. Sci. USA 31 (1945), 333–338.Google Scholar
Johnstone, P. T.. Sketches of an Elephant: a Topos Theory Compendium, volume 2 (Oxford University Press, Oxford, 2002).Google Scholar
Joyal, A. and Tierney, M.. An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. 51(309) (1984).Google Scholar
Manuell, G.. Quantalic spectra of semirings. PhD. thesis University of Edinburgh (2019).Google Scholar
Marsden, D.. Category theory using string diagrams (2014). arXiv preprint, arXiv:1401.7220.Google Scholar
Niefield, S. B.. Projectivity, continuity, and adjointness: quantales, Q-posets and Q-modules. Theory Appl. Categ. 31(30) (2016), 839–851.Google Scholar
Picado, J. and Pultr, A.. Frames and Locales: Topology without Points. Frontiers in Mathematics. (Springer, Basel, 2012).CrossRefGoogle Scholar
Pitts, A. M.. Notes on categorical logic, 1991. Unpublished lecture notes, https://www.cl.cam.ac.uk/amp12/papers/notcl/notcl.pdf.Google Scholar
Resende, P. and Vickers, S.. Localic suplattices and tropological systems. Theoret. Comput. Sci. 305(1-3) (2003), 311–346.CrossRefGoogle Scholar
Rosenthal, K. I.. Quantales and their applications. Pitman Research Notes in Mathematics Series vol. 234 (Longman Scientific and Technical, Harlow, 1990).Google Scholar
Selinger, P.. A survey of graphical languages for monoidal categories. In Coecke, B., editor, New Structures for Physics. Lecture Notes in Physics, vol. 813 (Springer, Berlin, 2010), pp. 289–355.Google Scholar
Stone, M. H.. Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41(3) (1937), 375–481.Google Scholar
Stone, M. H.. Topological representations of distributive lattices and Brouwerian logics. Časopis Pěst. Mat. Fys. 67(1) (1938), 1–25.CrossRefGoogle Scholar
Vickers, S. J. and Townsend, C. F.. A universal characterisation of the double powerlocale. Theoret. Comput. Sci. 316(1-3) (2004), 297–321.CrossRefGoogle Scholar