Skip to main content
Log in

The Płonka product of topological spaces

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We introduce a topological counterpart to the Płonka sums of algebraic structures: the Płonka product of topological spaces. This leads to a duality when considering spaces that are dually equivalent to the algebras used in the construction of the Płonka sum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  1. Bergman, C., Failing, D.: Commutative idempotent groupoids and the constraint satisfaction problem. Algebra Univers. 73, 391–417 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bochvar, D.: On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus. Mat. Sb. 4, 287–308 (1938)

    MATH  Google Scholar 

  3. Bonzio, S.: Dualities for Płonka sums. Log. Univers. 12, 327–339 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bonzio, S., Gil-Férez, J., Paoli, F., Peruzzi, L.: On Paraconsistent Weak Kleene Logic: axiomatization and algebraic analysis. Stud. Log. 105, 253–297 (2017)

    Article  Google Scholar 

  5. Bonzio, S., Loi, A., Peruzzi, L.: A duality for involutive bisemilattices. Stud. Log. 107, 423–444 (2019)

    Article  MathSciNet  Google Scholar 

  6. Bonzio, S., Moraschini, T., Pra Baldi, M.: Logics of left variable inclusion and Płonka sums of matrices. Submitted manuscript (2018). Preprint https://arxiv.org/pdf/1804.08897.pdf

  7. Bonzio, S., Pra Baldi, M.: Containment logics and Płonka sums of matrices. Submitted manuscript (2018). Preprint arxiv:1809.06761

  8. Brümmer, G.: Topological categories. Topol. Appl. 18, 27–41 (1984)

    Article  MathSciNet  Google Scholar 

  9. Ciuni, R., Ferguson, T., Szmuc, D.: Logics based on linear orders of contaminating values. Submitted manuscript (2018)

  10. Clifford, A., Preston, G.: The Algebraic Theory of Semigroups. Mathematical Surveys and Monographs. American Mathematical Society, London (1961)

    Google Scholar 

  11. Ferguson, T.: A computational interpretation of conceptivism. J. Appl. Non-Class. Log. 24, 333–367 (2014)

    Article  MathSciNet  Google Scholar 

  12. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  13. Gierz, G., Romanowska, A.: Duality for distributive bisemilattices. J. Aust. Math. Soc. A 51, 247–275 (1991)

    Article  MathSciNet  Google Scholar 

  14. Haimo, F.: Some limits of Boolean algebras. Proc. Am. Math. Soc. 2, 566–576 (1951)

    Article  MathSciNet  Google Scholar 

  15. Harding, J., Romanowska, A.: Varieties of Birkhoff systems: part I. Order 34, 45–68 (2017)

    Article  MathSciNet  Google Scholar 

  16. Harding, J., Romanowska, A.: Varieties of Birkhoff systems: part II. Order 34, 69–89 (2017)

    Article  MathSciNet  Google Scholar 

  17. Libkin, L.: Aspects of Partial Information in Databases. Ph.D. thesis, University of Pennsylvania (1994)

  18. Mardešić, S., Segal, J.: Shape Theory: The Inverse System Approach. North-Holland Mathematical Library, North-Holland (1982)

    MATH  Google Scholar 

  19. Płonka, J.: On a method of construction of abstract algebras. Fund. Math. 61, 183–189 (1967)

    Article  MathSciNet  Google Scholar 

  20. Płonka, J.: On distributive quasilattices. Fund. Math. 60, 191–200 (1967)

    Article  MathSciNet  Google Scholar 

  21. Płonka, J.: On sums of direct systems of Boolean algebras. Colloq. Math. 20, 209–214 (1969)

    Article  MathSciNet  Google Scholar 

  22. Płonka, J.: On the sum of a direct system of universal algebras with nullary polynomials. Algebra Universalis 19(2), 197–207 (1984)

    Article  MathSciNet  Google Scholar 

  23. Płonka, J., Romanowska, A.: Semilattice sums. In: Romanowska, A., Smith, J. (eds.) Universal Algebra and Quasigroup Theory. Research and Exposition in Mathematics, vol. 19, pp. 123–158. Springer, New York (1992)

    Google Scholar 

  24. Puhlmann, H.: The snack powerdomain for database semantics. In: Borzyszkowski, A., Sokołowski, S. (eds.) Mathematical Foundations of Computer Science, pp. 650–659. Springer, Berlin (1993)

    Google Scholar 

  25. Romanowska, A., Smith, J.: Semilattice-based dualities. Stud. Log. 56, 225–261 (1996)

    Article  MathSciNet  Google Scholar 

  26. Romanowska, A., Smith, J.: Duality for semilattice representations. J. Pure Appl. Algebra 115, 289–308 (1997)

    Article  MathSciNet  Google Scholar 

  27. Romanowska, A., Smith, J.: Modes. World Scientific, Singapore (2002)

    Book  Google Scholar 

  28. Singer, I., Thorpe, J.: Lecture Notes on Elementary Topology and Geometry. Undergraduate Texts in Mathematics. Springer, New York (1976)

    Book  Google Scholar 

  29. Stone, M.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)

    Article  MathSciNet  Google Scholar 

  30. Szmuc, D.: Defining LFIs and LFUs in extensions of infectious logics. J. Appl. Non-Class. Log. 26, 286–314 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stefano Bonzio.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of the Andrea Loi is supported by INdAM, GNSAGA—Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni and by GESTA—Funded by Fondazione di Sardegna and Regione Autonoma della Sardegna.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonzio, S., Loi, A. The Płonka product of topological spaces. Algebra Univers. 80, 29 (2019). https://doi.org/10.1007/s00012-019-0599-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-019-0599-z

Keywords

Mathematics Subject Classification

Navigation