algebra universalis

, Volume 14, Issue 1, pp 140–166 | Cite as

Construction of free continuous algebras

  • Jiří Adámek
Article

Abstract

A subset system Z is a function, assiging to each posetX a collectionZX of its subsets. We studyZ-continuous universal algebras, i.e., ordered algebras such that their domains have, and their operations preserve, joins of allZ-sets. A construction of the freeZ-continuous algebras is exhibited for an arbitraryZ. It turn out that only four subset systems have to be considered: the systemZX of (i) all ε-chains inX; (ii) all finite subsets ofX; (iii) all countable subsets ofX and (iv) the void set alone. The free algebras are described concretely for these four cases.

Keywords

Disjoint Union Binary Tree Universal Property Algebra UNIV Extension Property 

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References

  1. [1]
    J. Adámek,Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolinae15 (1974), 589–602.MATHMathSciNetGoogle Scholar
  2. [2]
    B. Banaschwski andE. Nelson,Completion of partially ordered sets as reflections. Computer Science Tech. Rep. 79-CS-6, McMaster University, Hamilton, Ontario, Canada, 1979.Google Scholar
  3. [3]
    G. Jarzembski,Algebry v kategorii zbiorow czesciovo uporiadkovanych. Preprint 1–1978, Mathematical Institute of N. Copernicus University, Torun, Poland, 1978.Google Scholar
  4. [4]
    E. Nelson,Z-continuous algebras. To appear.Google Scholar
  5. [5]
    D. Scott,Continuous lattices. Proceedings, 1971 Dalhouisie Conf. Springer Lecture Note Series, 274, Springer-Verlag (1972), 97–136.Google Scholar
  6. [6]
    E. G. Wagner, J. W. Thatcher andJ. B. Wright,A uniform approach to inductive posets and inductive closure. Lecture Notes in Computer Science 53 (MFCS 77 Proceedings), Springer-Verlag 1977.Google Scholar
  7. [7]
    E. G. Wagner, J. W. Thatcher andJ. B. Wright,Free continuous theories. IBM Research Rep. RC 6906, IBM Research Division, 1979.Google Scholar
  8. [8]
    J. Adámek, E. Nelson, Y. Reiterman, Tree constructions of free continuous algebras. To appear.Google Scholar

Copyright information

© Birkhäuser Verlag 1982

Authors and Affiliations

  • Jiří Adámek
    • 1
  1. 1.Faculty of Electrical EngineeringTechnical University PraguePrahaCzechoslovakia

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