On Final Coalgebras of Power-Set Functors and Saturated Trees To George Janelidze on the Occasion of His Sixtieth Birthday
On Final Coalgebras of Power-Set Functors and Saturated Trees To George Janelidze on the Occasion of His Sixtieth Birthday
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关于幂集函子和饱和树的最终余代数致乔治·贾内利泽 (George Janelidze) 六十岁生日之际
DOI：
10.1007/s10485-014-9372-9
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发表时间：
2014
影响因子：
0.6
通讯作者：
Adámek J
中科院分区：
文献类型：
--
作者：
Adámek J
Worrell描述了有限幂集函子的最终协代数，并证明了最终链以ω+ω步收敛。我们将步长ω描述为饱和树的集合，这个概念等价于K. Fine在20世纪70年代研究模态逻辑时引入的模态饱和树。对于有界幂集函子spλ，其中λ是一个无限正则基数，我们证明了其构造需要λ+ω阶跃。我们还推广了Worrell关于可交换一元的tom标记树的结果，得到了h . p .研究过的相应函子的最终协代数。Gumm和T. Schröder。通过对所有序数引入非饱和树的概念来描述幂集函子的最终链，并证明在非共性ω下，最终链的第1步由非饱和的强扩展树组成。
The final coalgebra for the finite power-set functor was described by Worrell who also proved that the final chain converges inω+ωsteps. We describe the stepωas 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 functorsPλ, whereλis an infinite regular cardinal, we prove that the construction needs preciselyλ+ωsteps. We also generalize Worrell’s result toM-labeled trees for a commutative monoidM, yielding a final coalgebra for the corresponding functorℳfstudied by H.-P. Gumm and T. Schröder. We describe the final chain of the power-set functor by introducing the concept ofi-saturated tree for all ordinalsi, and then prove that foriof cofinalityω, thei-th step in the final chain consists of alli-saturated, strongly extensional trees.