Applied Categorical Structures

, Volume 23, Issue 4, pp 609–641 | Cite as

On Final Coalgebras of Power-Set Functors and Saturated Trees

To George Janelidze on the Occasion of His Sixtieth Birthday
  • Jiří Adámek
  • Paul B. Levy
  • Stefan Milius
  • Lawrence S. Moss
  • Lurdes Sousa
Article

Abstract

The final coalgebra for the finite power-set functor was described by Worrell who also proved that the final chain converges in ω+ω steps. We describe the step ω as the set of saturated trees, a concept equivalent to the modally saturated trees introduced by K. Fine in the 1970s in his study of modal logic. And for the bounded power-set functors Pλ, where λ is an infinite regular cardinal, we prove that the construction needs precisely λ+ω steps. We also generalize Worrell’s result to M-labeled trees for a commutative monoid M, yielding a final coalgebra for the corresponding functor f studied by H.-P. Gumm and T. Schröder. We describe the final chain of the power-set functor by introducing the concept of i-saturated tree for all ordinals i, and then prove that for i of cofinality ω, the i-th step in the final chain consists of all i-saturated, strongly extensional trees.

Keywords

Saturated tree Extensional tree Final coalgebra Power-set functor Modal logic 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Jiří Adámek
    • 1
  • Paul B. Levy
    • 2
  • Stefan Milius
    • 3
  • Lawrence S. Moss
    • 4
  • Lurdes Sousa
    • 5
    • 6
  1. 1.Institut für Theoretische InformatikTechnische Universität BraunschweigBraunschweigGermany
  2. 2.School of Computer ScienceUniversity of BirminghamBirminghamUK
  3. 3.Lehrstuhl für Theoretische InformatikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  4. 4.Department of MathematicsIndiana UniversityBloomingtonUSA
  5. 5.Department of MathematicsPolytechnic Institute of ViseuViseuPortugal
  6. 6.Centre for Mathematics of the University of CoimbraCoimbraPortugal

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