Terminal coalgebras in well-founded set theory
Theoretical Computer Science
On the greatest fixed point of a set functor
Theoretical Computer Science
Universal coalgebra: a theory of systems
Theoretical Computer Science - Modern algebra and its applications
Automata and Algebras in Categories
Automata and Algebras in Categories
On the final sequence of a finitary set functor
Theoretical Computer Science
Completely iterative algebras and completely iterative monads
Information and Computation
Connections of coalgebra and semantic modeling
Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge
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We prove that the iterative construction of initial algebras converges for endofunctors F of many-sorted sets whenever F has an initial algebra. In the case of one-sorted sets, the convergence takes n steps where n is either an infinite regular cardinal or is at most 3. Dually, the existence of a many-sorted terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. Moreover, every endofunctor with a fixed-point pair larger than the number of sorts is proved to have a terminal coalgebra. As demonstrated by James Worell, the number of steps here need not be a cardinal even in the case of a single sort: it is ?? + ?? for the finite power-set functor. The above results do not hold for related categories, such as graphs: we present non-constructive initial algebras and terminal coalgebras.