New results on hierarchies of domains

  • Achim Jung
Part III Domain Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 298)


The relation between a cpo D and its space [DD] of Scott-continuous functions is investigated. For a cpo D with a least element we show that if [DD] is algebraic then D itself is algebraic. Together with a generalization of Smyth's Theorem to strict functions this implies that [DD] is ω-algebraic whenever \([D \xrightarrow{s} D]\) is. It is an open question, whether [DD] is ω-algebraic whenever \([D \xrightarrow{s} D]\) is algebraic.

Smyth's Theorem is extended to the class of cpo's with no least element assumed. Our main result asserts that in this context the profinites again form the largest cartesian closed category of domains. The proof also yields the following: if the functionspace of a cpo D is algebraic, then D has infima for filtered sets. The question, whether an ω-algebraic functionspace implies that D is profinite, remains open.


Function Space Minimal Element Compact Element Compact Function Infinite Subset 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Gunter:Profinite Solutions for Recursive Domain Equations. Doctoral Dissertation, University of Wisconsin, Madison 1985.Google Scholar
  2. [2]
    C. Gunter:Comparing Categories of Domains. Technical report, Carnegie-Mellon University, Pittsburg 1985.Google Scholar
  3. [3]
    G. Markowsky:Chain-complete p.o. sets and directed sets with applications. Algebra Universalis 6 (1976), p.53–68.Google Scholar
  4. [4]
    G. Plotkin:A powerdomain construction. SIAM J. Comput. 5 (1976), p.452–488.CrossRefGoogle Scholar
  5. [5]
    M.B. Smyth:The Largest Cartesian Closed Category of Domains. Theoretical Computer Science 27 (1983), p.109–119.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Achim Jung
    • 1
  1. 1.Fachbereich MathematikTechnische Hochschule DarmstadtDarmstadtWest Germany

Personalised recommendations