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

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**On Final Coalgebras of Power-Set Functors and Saturated Trees : To George Janelidze on the Occasion of His Sixtieth Birthday.** / Adámek, Jiří; Levy, Paul B.; Milius, Stefan; Moss, Lawrence S.; Sousa, Lurdes.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Applied Categorical Structures*, vol. 23, no. 4, pp. 609-641. https://doi.org/10.1007/s10485-014-9372-9

## APA

*Applied Categorical Structures*,

*23*(4), 609-641. https://doi.org/10.1007/s10485-014-9372-9

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## Bibtex

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## RIS

TY - JOUR

T1 - On Final Coalgebras of Power-Set Functors and Saturated Trees

T2 - To George Janelidze on the Occasion of His Sixtieth Birthday

AU - Adámek, Jiří

AU - Levy, Paul B.

AU - Milius, Stefan

AU - Moss, Lawrence S.

AU - Sousa, Lurdes

PY - 2015/8/1

Y1 - 2015/8/1

N2 - 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.

AB - 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.

KW - Saturated tree

KW - Extensional tree

KW - Final coalgebra

KW - Power-set functor

KW - Modal logic

U2 - 10.1007/s10485-014-9372-9

DO - 10.1007/s10485-014-9372-9

M3 - Article

VL - 23

SP - 609

EP - 641

JO - Applied Categorical Structures

JF - Applied Categorical Structures

SN - 0927-2852

IS - 4

ER -