Corecursive Algebras, Corecursive Monads and Bloom Monads

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras generalizing completely iterative algebras. We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras

On Free Completely Iterative Algebras

For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely iterative algebra. Moreover, the algebra structure of the latter is the unique continuous extension of the algebra structure of the free algebra. For general finitary functors the free algebra and the free completely iterative algebra are proved to be posets sharing the same conservative completion. And for every recursive equation in the free completely iterative algebra the solution is obtained as the join of an ?-chain of approximate solutions in the free algebra

How Iterative are Iterative Algebras?

AbstractIterative algebras are defined by the property that every guarded system of recursive equations has a unique solution. We prove that they have a much stronger property: every system of recursive equations has a unique strict solution. And we characterize those systems that have a unique solution in every iterative algebra