Well-Pointed Coalgebras (Extended Abstract)

  • Jiří Adámek
  • Stefan Milius
  • Lawrence S. Moss
  • Lurdes Sousa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7213)


For set functors preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. And the initial algebra consists of all well-pointed coalgebras that are well-founded in the sense of Taylor [16]. Finally, the initial iterative algebra consists of all finite well-pointed coalgebras. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems.


Binary Tree Inverse Image Full Subcategory Regular Language Label Transition System 
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.


  1. 1.
    Aczel, P.: Non-well-founded Sets. CSLL Lect. Notes, vol. 14. Stanford CSLI Publications, Stanford (1988)zbMATHGoogle Scholar
  2. 2.
    Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolinæ 15, 589–602 (1974)zbMATHGoogle Scholar
  3. 3.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. John Wiley and Sons, New York (1990)zbMATHGoogle Scholar
  4. 4.
    Adámek, J., Milius, S., Velebil, J.: Iterative algebras at work. Math. Structures Comput. Sci. 16, 1085–1131 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Cambridge University Press (1994)Google Scholar
  6. 6.
    Capretta, V., Uustalu, T., Vene, V.: Recursive coalgebras from comonads. Inform. and Comput. 204, 437–468 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien. Lecture Notes in Math., vol. 221. Springer, Berlin (1971)zbMATHGoogle Scholar
  8. 8.
    Gumm, H.-P.: On minimal coalgebras. Appl. Categ. Structures 16, 313–332 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Manes, E.G., Arbib, M.A.: Algebraic Approaches to Program Semantics. Springer, New York (1986)zbMATHGoogle Scholar
  10. 10.
    Milius, S.: A sound and complete calculus for finite stream circuits. In: Proc. 25th Annual Symposium on Logic in Computer Science (LICS 2010), pp. 449–458. IEEE Computer Society (2010)Google Scholar
  11. 11.
    Nelson, E.: Iterative algebras. Theoret. Comput. Sci. 25, 67–94 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Osius, G.: Categorical set theory: a characterization of the category of sets. J. Pure Appl. Algebra 4, 79–119 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoret. Comput. Sci. 249, 3–80 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rutten, J.J.M.M., Turi, D.: On the Foundations of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) REX 1992. LNCS, vol. 666, pp. 477–530. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  15. 15.
    Tarski, A.: A lattice theoretical fixed point theorem and its applications. Pacific J. Math. 5, 285–309 (1955)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Taylor, P.: Towards a unified treatement of induction I: the general recursion theorem, preprint (1995–6),
  17. 17.
    Taylor, P.: Practical Foundations of Mathematics. Cambridge University Press (1999)Google Scholar
  18. 18.
    Tiurin, J.: Unique fixed points vs. least fixed points. Theoret. Comput. Sci. 12, 229–254 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Trnková, V.: On a descriptive classification of set functor I. Comment. Math. Univ. Carolinæ 12, 323–352 (1971)Google Scholar
  20. 20.
    Trnková, V., Adámek, J., Koubek, V., Reiterman, J.: Free algebras, input processes and free monads. Comment. Math. Univ. Carolinæ 16, 339–351 (1975)zbMATHGoogle Scholar
  21. 21.
    Worrell, J.: On the final sequence of a finitary set functor. Theoret. Comput. Sci. 338, 184–199 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jiří Adámek
    • 1
  • Stefan Milius
    • 1
  • Lawrence S. Moss
    • 2
  • Lurdes Sousa
    • 3
  1. 1.Institut für Theoretische InformatikTechnische Universität BraunschweigGermany
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Departamento de MatemáticaInstituto Politécnico de ViseuPortugal

Personalised recommendations