53
Views
0
CrossRef citations to date
0
Altmetric
Articles

A set-theoretic proof of the representation of MV-algebras by sheaves

ORCID Icon & ORCID Icon
Pages 317-334 | Received 01 Sep 2022, Accepted 08 Mar 2023, Published online: 14 Apr 2023

References

  • Castiglioni, J., Menni, M., & Zuluaga Botero, W. (2016). A representation theorem for integral rigs and its applications to residuated lattices. Journal of Pure and Applied Algebra, 220(10), 3533–3566. https://doi.org/10.1016/j.jpaa.2016.04.014
  • Chang, C. C. (1959). A new proof of the completeness of the Łukasiewicz axioms. Transactions of the American Mathematical Society, 93(1), 74–80. https://doi.org/10.2307/1993423
  • Cignoli, R., Dubuc, E. J., & Mundici, D. (2004). Extending stone duality to multisets and locally finite MV-algebras. Journal of Pure and Applied Algebra, 189(1–3), 37–59. https://doi.org/10.1016/j.jpaa.2003.10.021
  • Cignoli, R. L. O., D'Ottaviano, I. M. L., & Mundici, D. (2000). Algebraic foundations of many-valued reasoning (Vol. 7). Kluwer Academic Publishers. doi:10.1007/978-94-015-9480-6
  • Comer, S. (1971). Representations by algebras of sections over boolean spaces. Pacific Journal of Mathematics, 38(1), 29–38. https://doi.org/10.2140/pjm
  • Coste, M. (1979). Localisation, spectra and sheaf representation. In Applications of sheaves Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977 (Vol. 753, pp. 212–238). Springer.
  • Dauns, J., & Hofmann, K. H. (1966). The representation of biregular rings by sheaves. Mathematische Zeitschrift, 91(2), 103–123. https://doi.org/10.1007/BF01110158
  • Davey, B. A. (1973). Sheaf spaces and sheaves of universal algebras. Mathematische Zeitschrift, 134(4), 275–290. https://doi.org/10.1007/BF01214692
  • Di Nola, A., Esposito, I., & Gerla, B. (2007). Local algebras in the representation of MV-algebras. Algebra Universalis, 56(2), 133–164. https://doi.org/10.1007/s00012-007-1984-6
  • Dubuc, E. J., & Poveda, Y. A. (2010). Representation theory of MV-algebras. Annals of Pure and Applied Logic, 161(8), 1024–1046. https://doi.org/10.1016/j.apal.2009.12.006
  • Dubuc, E. J., & Poveda, Y. A. (2015). On the equivalence between MV-algebras and l-groups with strong unit. Studia Logica, 103(4), 807–814. https://doi.org/10.1007/s11225-014-9593-9
  • Estrada, A., & Poveda, Y. A. (2019). MVW-rigs and product MV-algebras. Journal of Applied Non-Classical Logics, 29(1), 78–96. https://doi.org/10.1080/11663081.2018.1534795
  • Ferraioli, A. R., & Lettieri, A. (2011). Representations of MV-algebras by sheaves. MLQ. Mathematical Logic Quarterly, 57(1), 27–43. https://doi.org/10.1002/malq.200910116
  • Filipoiu, A., & Georgescu, G. (1995). Compact and pierce representations of MV-algebras. Revue Roumaine De Mathématiques Pures Et Appliquées. Romanian Journal of Pure and Applied Mathematics, 40(7-8), 599–618. https://www.researchgate.net/publication/265548706_Compact_and_Pierce_representations_of_MV-algebras
  • Gehrke, M. (2018). Sheaves and duality. Journal of Pure and Applied Algebra, 222(8), 2164–2180. https://doi.org/10.1016/j.jpaa.2017.09.004
  • Gehrke, M., van Gool, S. J., & Marra, V. (2014). Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality. Journal of Algebra, 417, 290–332. https://doi.org/10.1016/j.jalgebra.2014.06.031
  • Godement, R. (1958). Topologie algébrique et théorie des faisceaux. Hermann, Paris. Actualités Scientifiques et Industrielles. 1252. Publications de l'Institut de Mathématique de l'Université de Strasbourg. XIII. Hermann & Cie. viii, 283 p. (1958). Publ. Math. Univ. Strasbourg. No. 13.
  • Grothendieck, A., Dieudonné, J., & Dieudonné, J. (1971). Eléments de géométrie algébrique (Vol. 166). Springer. Retrieved from https://labs.thosgood.com/ega/book-auto.pdf
  • Hartshorne, R. (1977). Algebraic geometry (1st ed.). Springer. (No. 52). doi:10.1007/978-1-4757-3849-0
  • Johnstone, P. T. (1982). Stone spaces (Vol. 3). Cambridge University Press.
  • Johnstone, P. T. (1983). The point of pointless topology. Bulletin of the American Mathematical Society, 8(1), 41–53. https://doi.org/10.1090/bull/1983-08-01
  • Keimel, K. (1970). Darstellung von halbgruppen und universellen algebren durch schnitte in garben; bireguläre halbgruppen. Mathematische Nachrichten, 45(1–6), 81–96. https://doi.org/10.1002/(ISSN)1522-2616
  • Keimel, K. (1971). The representation of lattice-ordered groups and rings by sections in sheaves. In Lectures on the applications of sheaves to ring theory (pp. 1–98). Springer.
  • Leinster, T. (2014). Basic category theory. (Vol. 143), Cambridge University Press. doi:/10.1017/CBO9781107360068
  • Luan, W., Weber, H., & Yang, Y. (2021, February). Filter topologies and topological MV-algebras. Fuzzy Sets and Systems, 406, 11–21. https://doi.org/10.1016/j.fss.2020.08.017
  • MacLane, S. (1998). Categories for the working mathematician (4th corrected printing Ed., Vol. 5). Springer Science & Business Media.
  • MacLane, S., & Moerdijk, I. (1994). Sheaves in geometry and logic: A first introduction to topos theory. Springer.
  • Mulvey, C. J. (1978). Compact ringed spaces. Journal of Algebra, 52(2), 411–436. https://doi.org/10.1016/0021-8693(78)90248-X
  • Mulvey, C. J. (1979). Applications of sheaves. In Proc. res. symp. 1977 (pp. 542–585). Springer.
  • Mundici, D. (1986). Interpretation of AF C ∗-algebras in Łukasiewicz sentential calculus interpretation of AF C ∗-algebras in Łukasiewicz sentential calculus. Journal of Functional Analysis, 65(1), 15–63. https://doi.org/10.1016/0022-1236(86)90015-7
  • Nuñez, J. H. (2012). Algunas propiedades del espectro primo en MV-Algebras [Doctoral dissertation]. Technological University of Pereira).
  • Pierce, R. S. (1967). Modules over commutative regular rings (Vol. 70), American Mathematical Society. doi:10.1090/memo/0070
  • Poveda, Y. (2007). Una teoría general de representación para mv-álgebra. Universidad de Buenos Aires.
  • Schwartz, N. (2013). Sheaves of Abelian l-groups. Order, 30(2), 497–526. https://doi.org/10.1007/s11083-012-9258-0
  • Vaggione, D. J. (1992). Sheaf representation and Chinese remainder theorems. Algebra Universalis, 29(2), 232–272. https://doi.org/10.1007/BF01190609
  • Wolf, A. (1974). Sheaf representations of arithmetical algebras. In Recent advances in the representation theory of rings and *-algebras by continuous sections (Vol. 148, pp. 87–93). Memoirs of the American Mathematical Society.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.