On Final Coalgebras of Power-Set Functors and Saturated Trees: To George Janelidze on the Occasion of His Sixtieth Birthday

Research output: Contribution to journalArticlepeer-review


  • Jiří Adámek
  • Paul Levy
  • Stefan Milius
  • Lawrence S. Moss
  • Lurdes Sousa

Colleges, School and Institutes


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 P λ , 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 ℳ 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.


Original languageEnglish
Pages (from-to)609-641
JournalApplied Categorical Structures
Issue number4
Early online date18 Jun 2015
Publication statusPublished - 1 Aug 2015


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