Abstract
In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that Archimedean almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters.
Article PDF
Similar content being viewed by others
References
Buskes, G., de Pagter, B., van Rooij, A.: Functional calculus on Riesz spaces. Indag. Math. (N.S.) 2(4), 423–436 (1991)
Birkhoff, G.: Lattice ordered groups. Ann. Math. 43(2), 298–331 (1942)
Banaschewski, B., Mulvey, C.J.: A constructive proof of the Stone-Weierstrass theorem. J. Pure Appl. Algebra, 116(1–3), 25–40 (1997). Special volume on the occasion of the 60th birthday of Professor Peter J. Freyd
Banaschewski, B., Mulvey, C.J.: The spectral theory of commutative C *-algebras: the constructive Gelfand-Mazur theorem. Quaest. Math. 23(4), 465–488 (2000)
Banaschewski, B., Mulvey, C.J.: The spectral theory of commutative C *-algebras: the constructive spectrum. Quaest. Math. 23(4), 425–464 (2000)
Banaschewski, B., Mulvey, C.J.: A globalisation of the Gelfand duality theorem. Ann. Pure Appl. Logic 137(1–3), 62–103 (2006)
Buskes, G., van Rooij, A.: Small Riesz spaces. Math. Proc. Camb. Philos. Soc. 105(3), 523–536 (1989)
Buskes, G., van Rooij, A.: Almost f-algebras: commutativity and the Cauchy-Schwarz inequality. Positivity 4(3), 227–231 (2000). Positivity and its applications (Ankara, 1998)
Cederquist, J., Coquand, T.: Entailment relations and distributive lattices. In: Logic Colloquium ’98 (Prague). Lect. Notes Log., vol. 13, pp. 127–139. Assoc. Symbol. Logic, Urbana, IL (2000)
Caspers, M., Heunen, C., Landsman, K., Spitters, B.: Intuitionistic quantum logic of an n-level system. Found. Phys. 39(7), 731–759 (2009)
Coquand, T.: Compact spaces and distributive lattices. J. Pure Appl. Algebra 184(1), 1–6 (2003)
Coquand, T.: About Stone’s notion of spectrum. J. Pure Appl. Algebra 197, 141–158 (2005)
Coquand, T., Spitters, B.: Formal topology and constructive mathematics: the Gelfand and Stone-Yosida representation theorems. J. Univers. Comput. Sci. 11(12), 1932–1944 (2005)
Coquand, T., Spitters, B.: A constructive proof of Gelfand duality for C*-algebras. Math. Proc. Camb. Philos. Soc. 147(2), 323–337 (2009). doi:10.1017/S0305004109002539
Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)
Heunen, C., Landsman, K., Spitters, B.: A topos for algebraic quantum theory. Commun. Math. Phys. 291(1), 63–110 (2009)
Heunen, C., Landsman, N.P., Spitters, B.: Bohrification of Operator Algebras and Quantum Logic (2009)
Johnstone, P.T.: Sketches of an Elephant: a Topos Theory Compendium, vol. 2. Clarendon Press (2002)
Lorenzen, P.: Abstrakte Begründung der multiplikativen Idealtheorie. Math. Z. 45, 533–553 (1939)
Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. I. North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1971)
Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. Universitext. Springer-Verlag, New York (1994) A first introduction to topos theory. Corrected reprint of the 1992 edition
Mulvey, C.J.: On the geometry of choice. In: Topological and Algebraic Structures in Fuzzy Sets. Trends Log. Stud. Log. Libr., vol. 20, pp. 309–336. Kluwer Acad. Publ., Dordrecht (2003)
Riesz, F.: Sur quelques notions fondamentales dans la théorie générale des opérations linéaires. Ann. Math. 41(2), 174–206 (1940)
Sambin, G.: Intuitionistic formal spaces - a first communication. In: Skordev, D. (ed.) Mathematical Logic and Its Applications, pp. 187–204. Plenum (1987)
Stone, M.H.: A general theory of spectra. II. Proc. Nat. Acad. Sci. U. S. A. 27, 83–87 (1941)
Takeuti, G.: Two applications of logic to mathematics. In: Kanô Memorial Lectures, vol. 3. Publications of the Mathematical Society of Japan, No. 13. Iwanami Shoten, Publishers, Tokyo (1978)
Vickers, S.: Locales and toposes as spaces. In: Aiello, M., Pratt-Hartmann, I.E., van Benthem, J.F.A.K. (eds.) Handbook of Spatial Logics, chapter 8. Springer (2007)
Yosida, K.: On the representation of the vector lattice. Proc. Imp. Acad. Tokyo 18, 339–342 (1942)
Zaanen, A.C.: Riesz spaces. II. In: North-Holland Mathematical Library, vol. 30. North-Holland Publishing Co., Amsterdam (1983)
Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer-Verlag, Berlin (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Spitters, B. Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory. Order 27, 225–233 (2010). https://doi.org/10.1007/s11083-010-9147-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-010-9147-3