Skip to main content

Propaedeutic to chain-complete posets with basis

Part of the Lecture Notes in Mathematics book series (LNM,volume 871)

Abstract

In this paper, several relations between members of a chain-complete poset (weaker than the relations relatively compact and IN introduced by Scott in his study of bases) are considered, with an eye toward examining whether all the properties of bases can be obtained from weaker axioms. It turns out that for all but one relation (relatively chain compact) one gets a weaker notion of bases than using relatively compact or IN. For relatively chain compact we conjecture that one gets a weaker notion of basis. However, we show that in the presence of bounded joins the existence of a basis derived from the relation of relatively chain compact implies the existence of a Scott basis if the cardinality of the chain-complete poset is less than the ω-th infinite cardinal. If one accepts the Continuum Hypothesis this means that the weaker relation is adequate in many posets likely to be of interest to computer science. Finally, even in the presence of the "weaker" type of basis in a complete lattice, the meet operation is continuous.

Keywords

  • Local Basis
  • Complete Lattice
  • Continuum Hypothesis
  • Continuous Lattice
  • Weak Notion

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Astesiano, E., "Bases for Scott-continuous Posets," Universita di Genova, Pubblicazioni dell' Instituto di Matematica No. (181) 1976.

    Google Scholar 

  2. Egli, H. and Constable, R. L., "Computability Concepts for Programming Language Semantics," Theoretical Comp. Science, 2 (1976) 133–145.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Markowsky, G., "Chain-complete Posets and Directed Sets with Applications," Algebra Universalis 6 (1976) 53–68.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Markowsky, G., "A Motivation and Generalization of Scott's Notion of a Continuous Lattice," These Proceedings.

    Google Scholar 

  5. Markowsky, G. and Rosen, B. K., "Bases for Chain-complete Posets," IBM J. of Res. and Dev., 20 (March, 1976) 138–147.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Reynolds, J. "Notes on a Lattice-theoretic Approach to the Theory of Computation," Lecture Notes, Syracuse U., October, 1972.

    Google Scholar 

  7. Reynolds J., "On the Interpretation of Scott Domains" Symp. Math. 15 (1975) 123–135.

    Google Scholar 

  8. Scott, D., "Outline of a Mathematical Theory of Computation," Proc. 4th Ann. Princeton Conf. on Inform. Sci. and Systems, 1970, 169–176.

    Google Scholar 

  9. Scott, D., "Continuous Lattices," Proc. Dalhousie Conf. on Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer-Verlag, Berlin, 1972, 97–136.

    CrossRef  Google Scholar 

  10. Scott, D., "Data Types as Lattices," SIAM J. Comput. 5 (1976) 522–587.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Smyth, M., "Effectively Given Domains," Theoretical Comp. Science 5 (1977) 257–274.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Vuillemin, J., Syntaxe, Semantique, et Axiomatique d'un Language de Programmation Simple, These d'etat, University of Paris, Sept. 1974.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Markowsky, G. (1981). Propaedeutic to chain-complete posets with basis. In: Banaschewski, B., Hoffmann, RE. (eds) Continuous Lattices. Lecture Notes in Mathematics, vol 871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089914

Download citation

  • DOI: https://doi.org/10.1007/BFb0089914

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10848-1

  • Online ISBN: 978-3-540-38755-8

  • eBook Packages: Springer Book Archive