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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 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 language | English |
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Pages (from-to) | 609-641 |
Journal | Applied Categorical Structures |
Volume | 23 |
Issue number | 4 |
Early online date | 18 Jun 2015 |
DOIs | |
Publication status | Published - 1 Aug 2015 |
Keywords
- Saturated tree
- Extensional tree
- Final coalgebra
- Power-set functor
- Modal logic
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Dive into the research topics of 'On Final Coalgebras of Power-Set Functors and Saturated Trees: To George Janelidze on the Occasion of His Sixtieth Birthday'. Together they form a unique fingerprint.Projects
- 1 Finished
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Advanced Research Fellowship : Semantics of Non-determinism: Functions, Strategies and Bisimulation
Engineering & Physical Science Research Council
1/01/08 → 31/12/12
Project: Research Councils