# 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

## Standard

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.

In: Applied Categorical Structures, Vol. 23, No. 4, 01.08.2015, p. 609-641.

Research output: Contribution to journalArticlepeer-review

## Author

Adámek, Jiří ; Levy, Paul B. ; Milius, Stefan ; Moss, Lawrence S. ; Sousa, Lurdes. / On Final Coalgebras of Power-Set Functors and Saturated Trees : To George Janelidze on the Occasion of His Sixtieth Birthday. In: Applied Categorical Structures. 2015 ; Vol. 23, No. 4. pp. 609-641.

## Bibtex

@article{e4684e5d99ce43d5b692b8727eb32434,
title = "On Final Coalgebras of Power-Set Functors and Saturated Trees: To George Janelidze on the Occasion of His Sixtieth Birthday",
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{\textquoteright}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. Schr{\"o}der. 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",
author = "Ji{\v r}{\'i} Ad{\'a}mek and Levy, {Paul B.} and Stefan Milius and Moss, {Lawrence S.} and Lurdes Sousa",
year = "2015",
month = aug,
day = "1",
doi = "10.1007/s10485-014-9372-9",
language = "English",
volume = "23",
pages = "609--641",
journal = "Applied Categorical Structures",
issn = "0927-2852",
publisher = "Springer",
number = "4",

}

## 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 - 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 -