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Pointfree Functorial Polar Functions

Abstract

C denotes the category of compact regular frames with frame homomorphisms. A function \(\mathcal {X}\), which assigns to each C-object F a subalgebra of \(\mathcal {P}(F)\) that contains the complemented elements of F is said to be a polar function. An essential extension H of F is a \(\mathcal {X}\)-splitting frame of F if whenever \(p \in \mathcal {X}(F)\), then the polar generated by p in H is complemented. For F∈ C we examine the least \(\mathcal {X}\)-splitting extension and prove that every invariant polar function generates a C-hull class of frames. In addition, we define the concept of a functorial polar function and prove that each functorial polar function generates an epireflective subcategory of the category compact regular frames with skeletal maps.

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References

  1. Banaschewski, B: Compact regular frames and the Sikorski theorem. Kyungpook Math. Jour. 28(1), 1–14 (1988)

    MATH  MathSciNet  Google Scholar 

  2. Banaschewski, B., Pultr, A.: Booleanization, Cahiers Topologie Géom. Différentielle Catég. XXXVII-1, 41–60 (1996)

    MathSciNet  Google Scholar 

  3. Carrera, R.E.: Functorial Polar Functions. Math. Slovaca 61(3), 389–410 (2011)

    MATH  MathSciNet  Google Scholar 

  4. Hager, A.W., Martinez, J.: Patch-generated frames and projectable hulls. Appl. Categ. Struct. 15(1-2), 49–80 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  5. Herrlich, H., Strecker, G.: Category Theory. Sigma series in Pure Math No. 1. Heldermann Verlag, Berlin (1979)

    Google Scholar 

  6. Martinez, J.: Polar Functions, I: The summand-inducing hull of an archimedean -group with unit. In: Kluwer Acad. Pub. (ed.) In: Ord. Alg. Struc.: Proc. Gainesville Conf., 2001, pp 275–299. Kluwer Acad. Pub. (2002)

  7. Martinez, J., Zenk, E.R.: Nuclear typing of frames vs spatial selectors. Appl. Categ. Struct. 14(1), 35–61 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  8. Martinez, J., Zenk, E.R.: Epicompletion in frames with skeletal maps, I. Appl. Categ. Struct. 16(4), 521–533 (2008)

    MATH  MathSciNet  Article  Google Scholar 

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Correspondence to Ricardo E. Carrera.

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Carrera, R.E. Pointfree Functorial Polar Functions. Appl Categor Struct 24, 37–52 (2016). https://doi.org/10.1007/s10485-014-9385-4

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  • DOI: https://doi.org/10.1007/s10485-014-9385-4

Keywords

  • Pointfree polar functions
  • Functorial polar functions
  • Hull class
  • Skeletal maps
  • Compact regular frame
  • α-disconnected frame
  • α-cloz frame

Mathematics Subject Classification (2010)

  • 06D22
  • 18A99