Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T22:13:57.296Z Has data issue: false hasContentIssue false
  • English
  • Français

CHOICE-FREE STONE DUALITY

Published online by Cambridge University Press:  29 August 2019

NICK BEZHANISHVILI and
WESLEY H. HOLLIDAY
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM1090GE AMSTERDAM, THE NETHERLANDS E-mail:n.bezhanishvili@uva.nl
WESLEY H. HOLLIDAY
Affiliation:
DEPARTMENT OF PHILOSOPHY AND GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA94720, USA E-mail:wesholliday@berkeley.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Keywords

03G0506E1506D2203E25Stone dualityBoolean algebraregular open algebraVietoris hyperspacespectral spacesStone localesaxiom of choice

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020, pp. 109 - 148
Copyright
Copyright © The Association for Symbolic Logic 2019 

References

van Benthem, J., Possible worlds semantics for classical logic, Technical report ZW-8018, Department of Mathematics, Rijksuniversiteit, Groningen, December 1981.Google Scholar
van Benthem, J., Partiality and nonmonotonicity in classical logic. Logique et Analyse, vol. 29 (1986), no. 114, pp. 225–247.Google Scholar
van Benthem, J., A Manual of Intensional Logic, 2nd revised and expanded ed., CSLI Publications, Stanford, 1988.Google Scholar
van Benthem, J., Tales from an old manuscript, ILLC Prepublication Series PP-2016-30, 2016.Google Scholar
Bezhanishvili, G., Frames, topologies, and duality theory, Lecture Slides for TACL Summer School, June 15–19, 2015, http://logica.dmi.unisa.it/tacl/wp-content/uploads/2015/05/Salerno1.pdf.Google Scholar
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., and Kurz, A., Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, vol. 20 (2010), no. 3, pp. 359–393.Google Scholar
Carnahan, D., Some properties related to compactness in topological spaces, Ph.D. thesis, University of Arkansas, 1973.Google Scholar
Chagrov, A. and Zakharyaschev, M., Modal Logic, Clarendon Press, Oxford, 1997.Google Scholar
Esakia, L., Heyting Algebras: Duality Theory, Trends in Logic, Springer, Dordrecht, 2019 (English translation of 1985 Russian monograph).CrossRefGoogle Scholar
Gehrke, M. and Harding, J., Bounded lattice expansions. Journal of Algebra, vol. 238 (2001), no. 1, pp. 345–371.CrossRefGoogle Scholar
Givant, S. and Halmos, P., Introduction to Boolean Algebras, Springer, New York, 2009.Google Scholar
Goldblatt, R. I., The Stone space of an ortholattice. Bulletin of the London Mathematical Society, vol. 7 (1975), no. 1, pp. 45–48.Google Scholar
Halmos, P. R., Lectures on Boolean Algebras, D. Van Nostrand, Princeton, 1963.Google Scholar
Herrlich, H., Axiom of Choice, Lecture Notes in Mathematics, vol. 76, Springer, Berlin, 2006.Google Scholar
Hochster, M., Prime ideal structure in commutative rings. Transactions of the American Mathematical Society, vol. 142 (1969), pp. 43–60.Google Scholar
Holliday, W. H., Possibility frames and forcing for modal logic, UC Berkeley Working Paper in Logic and the Methodology of Science, 2015, http://escholarship.org/uc/item/5462j5b6.Google Scholar
Holliday, W. H., Possibility frames and forcing for modal logic (February 2018), UC Berkeley Working Paper in Logic and the Methodology of Science, 2018, https://escholarship.org/uc/item/0tm6b30q.Google Scholar
Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
Humberstone, L., From worlds to possibilities. Journal of Philosophical Logic, vol. 10 (1981), no. 3, pp. 313–339.Google Scholar
Johnstone, P. T., Tychonoff’s theorem without the axiom of choice. Fundamenta Mathematicae, vol. 112 (1981), no. 1, pp. 21–35.Google Scholar
Johnstone, P. T., Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982.Google Scholar
Johnstone, P. T., The point of pointless topology. Bulletin of the American Mathematical Society, vol. 8 (1983), no. 1, pp. 41–53.CrossRefGoogle Scholar
Massas, G., Possibility spaces, Q-completions and Rasiowa-Sikorski lemmas for non-classical logics, Master’s thesis, University of Amsterdam, 2016, ILLC Master of Logic Thesis.Google Scholar
Moshier, M. A. and Jipsen, P., Topological duality and lattice expansions, I: A topological construction of canonical extensions. Algebra Universalis, vol. 71 (2014), no. 2, pp. 109–126.CrossRefGoogle Scholar
Picado, J. and Pultr, A., Frames and Locales: Topology without Points, Frontiers in Mathematics, Birkhäuser, Basel, 2012.CrossRefGoogle Scholar
Plotkin, J. M., ZF and Boolean algebra. Israel Journal of Mathematics, vol. 23 (1976), no. 2–3, pp. 298–308.CrossRefGoogle Scholar
Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society, vol. 2 (1970), no. 1, pp. 196–190.CrossRefGoogle Scholar
Röper, P., Intervals and tenses. Journal of Philosophical Logic, vol. 9 (1980), no. 4, pp. 451–469.CrossRefGoogle Scholar
Schechter, E., Handbook of Analysis and its Foundations, Academic Press, New York, 1996.Google Scholar
Stone, M. H., The theory of representation for Boolean algebras. Transactions of the American Mathematical Society, vol. 40 (1936), no. 1, pp. 37–111.Google Scholar
Stone, M. H., Topological representation of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky, vol. 67 (1937), no. 1, pp. 1–25.Google Scholar
Tarski, A., Ideale in den Mengenkörpen. Annales de la Société Polonaise de Mathématique, vol. 15 (1937), pp. 186–189.Google Scholar
Tarski, A., Über additive und multiplikative Mengenkörper und Mengenfunktionen. Sprawozdania z Posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział III, Nauk Matematyczno-fizycznych, vol. 30 (1937), pp. 151–181.Google Scholar
Tarski, A., Der Aussagenkalkuül und die Topologie. Fundamenta Mathematicae, vol. 31 (1938), no. 1, pp. 103–134, English translation as Chapter 17 of [34].CrossRefGoogle Scholar
Tarski, A., Logic, Semantics, Metamathematics: Papers from 1923 to 1938, Clarendon Press, Oxford, 1956, Translated by J. H. Woodger.Google Scholar
Vickers, S., Topology via Logic, Cambridge University Press, Cambridge, 1989.Google Scholar
Tarski, A., Constructive points of powerlocales. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 122 (1997), no. 2, pp. 207–222.Google Scholar
Tarski, A., Localic completion of generalized metric spaces II: Powerlocales. Journal of Logic & Analysis, vol. 1 (2009), pp. 1–48.Google Scholar
Vietoris, L., Bereiche zweiter ordnung. Monatshefte für Mathematik und Physik, vol. 32 (1922), no. 1, pp. 258–280.CrossRefGoogle Scholar