Boolean Perspectives of Idioms and the Boyle Derivative


We are concerned with the boolean or more generally with the complemented properties of idioms (complete upper-continuous modular lattices). Simmons (Cantor–Bendixson, socle, and atomicity., 2014) introduces a device which captures in some informal speaking how far the idiom is from being complemented, this device is the Cantor-Bendixson derivative. There exists another device that captures some boolean properties, the so-called Boyle-derivative, this derivative is an operator on the assembly (the frame of nuclei) of the idiom. The Boyle-derivative has its origins in module theory. In this investigation we produce an idiomatic analysis of the boolean properties of any idiom using the Boyle-derivative and we give conditions on a nucleus j such that [jtp] is a complete boolean algebra. We also explore some properties of nuclei j such that \(A_{j}\) is a complemented idiom.

This is a preview of subscription content, access via your institution.


  1. 1.

    Arroyo Paniagua, M.J., Montes, J.R.: Some aspects of spectral torsion theories. Commun. Algebra 22(12), 4991–5003 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Boyle, A.K.: The large condition for rings with krull dimension. Proc. Am. Math. Soc. 72(1), 27–32 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Castro, J., González, P.M., Montes, J.: Some aspects of-full modules. East-West J. Math. 9(2), 139–159 (2007)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Golan, J.S.: Torsion Theories, vol. 29. Longman Scientific & Technical, Harlow (1986)

    Google Scholar 

  5. 5.

    González, P.M.: Algunos aspectos sobre módulos \(\tau \) -plenos, Ph.D. thesis, Universidad Nacional Autónoma de México (2008)

  6. 6.

    Gómez Pardo, J.L.: Spectral Gabriel topologies and relative singular functors. Commun. Algebra 13(1), 21–57 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Golan, J.S., Simmons, H.: Derivatives, nuclei and dimensions on the frame of torsion theories. Pitman Research Notes in Mathematics Series (1988)

  8. 8.

    Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  9. 9.

    José, M., Paniagua, A., Montes, J.R., Wisbauer, R.: Spectral torsion theories in module categories. Commun. Algebra 25(7), 2249–2270 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Jan-Erik, R.: Locally Distributive Spectral Categories and Strongly Regular Rings, Reports of the Midwest Category Seminar, pp. 156–181. Springer, Berlin (1967)

    Google Scholar 

  11. 11.

    Simmons, H.: The semiring of topologizing filters of a ring. Isr. J. Math. 61(3), 271–284 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Simmons, Harold: Near-discreteness of modules and spaces as measured by Gabriel and Cantor. J. Pure Appl. Algebra 56(2), 119–162 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Simmons, H.: The assembly of a frame. (2006)

  14. 14.

    Simmons, H.: The basics of frame theory. (2006)

  15. 15.

    Simmons, H.: The higher level CB properties of frames. (2006)

  16. 16.

    Simmons, H.: A decomposition theory for complete modular meet-continuous lattices. Algebra Universalis 64(3–4), 349–377 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Simmons, H.: How to generate G-topologies for module presheaf categories. (2012)

  18. 18.

    Simmons, H.: Cantor–Bendixson Properties of the Assembly of a Frame, Leo Esakia on Duality in Modal and Intuitionistic Logics, pp. 217–255. Springer, Berlin (2014)

    Google Scholar 

  19. 19.

    Simmons, H.: Cantor–Bendixson, socle, and atomicity., 2 (2014)

  20. 20.

    Simmons, H.: The Gabriel and the Boyle derivatives for a modular idiom., 3 (2014)

  21. 21.

    Simmons, H.: An introduction to idioms., 2 (2014)

  22. 22.

    Simmons, H.: A lattice theoretic analysis of a result due to Hopkins and Levitzki., 3 (2014)

  23. 23.

    Simmons, H.: The relative basic derivatives for an idiom., 2 (2014)

  24. 24.

    Simmons, H.: Examples of higher level assemblies (to appear) (2017)

  25. 25.

    Stenström, B.: Rings of Quotients: An Introduction to Methods of Ring Theory, vol. 217. Springer, Berlin (1975)

    Google Scholar 

  26. 26.

    Wilson, J.T.: The assembly tower and some categorical and algebraic aspects of frame theory. Ph.D. thesis, Carnegie Mellon University (1994)

Download references


We would like to thank the referee for a careful and detailed reading of the manuscript and suggestions to improve it. This work was supported by the grant UNAM-DGAPA-PAPIIT IN100517.

Author information



Corresponding author

Correspondence to Angel Zaldívar Corichi.

Additional information

This work was supported by the Grant UNAM-DGAPA-PAPIIT IN100517.

Communicated by Jorge Picado.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pérez, J.C., Bárcenas, M.M., Montes, J.R. et al. Boolean Perspectives of Idioms and the Boyle Derivative. Appl Categor Struct 27, 65–84 (2019).

Download citation


  • Complete boolean algebras
  • Lattices
  • Frames
  • Modules
  • Rings

Mathematics Subject Classification

  • Primary 06Cxx
  • Secondary 16S90