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The category of complete Boolean algebras is not an intersection of reflective subcategories of the category of frames

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Abstract

We answer a question of J. Rosický and W. Tholen by showing that the class of complete Boolean algebras (which is a prereflective subcategory of the category of frames) is not an intersection of reflective subcategories of the category of frames. In order to deduce this result, we first prove the following observation (using some basic facts about congruence frames): the three-element chain belongs to every reflective subcategory of the category of frames which contains the class of complete Boolean algebras.

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References

  1. 1.

    J. Adámek and J. Rosicky: On injectivity classes in locally presented categories, preprint, 1990 (to appear inTrans. Amer. Math. Soc.).

  2. 2.

    J. Adámek, H. Herrlich, and G. E. Strecker;Abstract and Concrete Categories, John Wiley and Sons, 1990.

  3. 3.

    B. Banaschewski: Unpublished seminar notes on Frames, University of Cape Town, 1988.

  4. 4.

    B. Banaschewski, J. L. Frith, and C. R. A. Gilmour: On the congruence lattice of a frame,Pacific J. Math. 130 (1987), 209–213.

  5. 5.

    J. L. Frith: Structured Frames, Ph.D. Thesis, University of Cape Town, 1987.

  6. 6.

    P. J. Freyd and G. M. Kelly: Categories of continuous functors,I,J. Pure Applied Algebra 2 (1972), 169–191.

  7. 7.

    H. Herrlich: Almost reflective subcategories of Top, preprint, 1991 (to appear inTopology and its Applications).

  8. 8.

    P. T. Johnstone:Stone Spaces, Cambridge University Press, 1982.

  9. 9.

    J. Rosický and W. Tholen: Orthogonal and prereflective subcategories,Cahiers Topologie Géom. Différentielle Catégoriques 29 (1988), 203–215.

  10. 10.

    W. Tholen: Prereflections and reflections,Comm. Algebra 14 (1986), 717–740.

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Additional information

Supported by the Categorical Topology Research Group at the University of Cape Town, under funding from the Foundation for Research Development and the University of Cape Town.

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Vajner, V. The category of complete Boolean algebras is not an intersection of reflective subcategories of the category of frames. Appl Categor Struct 1, 191–195 (1993). https://doi.org/10.1007/BF00880043

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Key words

  • Reflective subcategory
  • frame
  • complete Boolean algebra

Mathematics Subject Classifications (1991)

  • 18A40
  • 06E10