Inductive Construction of Repletion | Applied Categorical Structures
Skip to main content
Springer Nature Link
Log in

Inductive Construction of Repletion

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

The aim of this paper is to give a purely logical construction of repletion, i.e. the reflection of an arbitrary set to a replete one. Replete sets within constructive logic were introduced independently by M. Hyland and P. Taylor as the most restrictive but sufficiently general notion of predomain suitable for the purposes of denotational semantics à la Scott.

For any set A its repletion R(A) appears as an inductively defined subset of S2(A) ≡(A → S) → S which can be expressed within the internal language of a model of type theory. More explicitly, R(A) is the least subset of S2(A) containing all ‘point filters’ and closed under a class of ‘generalised limit processes’. Improvements of our construction arise from several results saying that it suffices for the purpose of repletion to consider more restrictive classes of ‘generalised limit processes’.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Freyd, P. et al.: Extensional pers, in Logic in Computer Science 5, IEEE, 1990.

  2. Hyland, J. M. E.: First steps towards synthetic domain theory, in Category Theory, Lecture Notes in Mathematics 1488, Springer, 1991, pp. 131–156.

  3. Hyland, M. and Moggi, E.: The S-replete construction, in CTCS 55, Springer Lecture Notes in Computer Science 953, 1995, pp. 96–116.

  4. Longley, J. R. and Simpson, A.: A uniform approach to domain theory in realisability models, Math. Structures Comput. Sci., 1995 to appear.

  5. Luo, Z. and Pollack, R.: Lego proof development system: User's manual, Technical Report ECS-LFCS–92–211, University of Edinburgh, 1992.

  6. McCarty, D.: Realizability and recursive mathematics, PhD Thesis, Carnegie-Mellon Univ., Pittsburgh, 1984.

    Google Scholar 

  7. Reus, B.: Program verification in synthetic domain theory, PhD Thesis, Ludwig-Maximilans-Univ., Munich, 1995.

    Google Scholar 

  8. Rosolini, G.: Notes on synthetic domain theory, Draft paper, Univ. of Genova, 1995.

  9. Streicher, T.: Semantics of Type Theory, Birkhäuser, Basel, 1991.

    Google Scholar 

  10. Streicher, T.: Dependence and independence results for (impredicative) calculi of dependent types, Math. Structures Comput. Sci. 2 (1992), 29–54.

    Google Scholar 

  11. Taylor, P.: The fixed point property in synthetic domain theory, in Logic in Computer Science 6, IEEE, 1991, pp. 152–160.

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics, TH Darmstadt, Schlossgartenstraße 7, D-64289, Darmstadt, Germany. e-mail

    Thomas Streicher

Authors
  1. Thomas Streicher
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Streicher, T. Inductive Construction of Repletion. Applied Categorical Structures 7, 185–207 (1999). https://doi.org/10.1023/A:1008604622568

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008604622568

Search

Navigation