Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 01 Oct 2020 08:58:31 GMT2020-10-01T08:58:31Z5011On Final Coalgebras of Power-Set Functors and Saturated Treeshttp://hdl.handle.net/10316/43810Title: On Final Coalgebras of Power-Set Functors and Saturated Trees
Authors: Adámek, Jiří; Levy, Paul B.; Milius, Stefan; Moss, Lawrence S.; Sousa, Lurdes
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.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/438102014-01-01T00:00:00Z